pdf2htmlEX/demo/cheat6.page

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class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">Theoretical<span class="_ _0"> </span>Computer<span class="_ _0"> </span>Science<span class="_ _0"> </span>Cheat<span class="_ _0"> </span>Sheet</div><div class="t m0 x116 h3 y2 ff3 fs0 fc0 sc0 ls0 ws0">π<span class="_ _6a"> </span><span class="ff2">Calculus</span></div><div class="t m0 x80 h3 y3 ff2 fs0 fc0 sc0 ls0 ws0">W<span class="_ _5"></span>allis’<span class="_ _7"> </span>identit<span class="_ _5"></span>y:</div><div class="t m0 x1a3 h4 y2f5 ff3 fs0 fc0 sc0 ls0 ws0">π <span class="ff2 ls1">=2<span class="_ _5"></span><span class="ff4 ls0">·</span></span></div><div class="t m0 xe5 h4 yd ff2 fs0 fc0 sc0 ls0 ws0">2<span class="_ _8"> </span><span class="ff4">·<span class="_ _8"> </span></span>2<span class="_ _8"> </span><span class="ff4">·<span class="_ _8"> </span></span>4<span class="_ _8"> </span><span class="ff4">·<span class="_ _8"> </span></span>4<span class="_ _8"> </span><span class="ff4">·<span class="_ _8"></span></span>6<span class="_ _8"> </span><span class="ff4">·<span class="_ _8"></span></span>6<span class="_ _6"> </span><span class="ff4 ls4">···</span></div><div class="t m0 xe5 h4 y431 ff2 fs0 fc0 sc0 ls0 ws0">1<span class="_ _8"> </span><span class="ff4">·<span class="_ _8"> </span></span>3<span class="_ _8"> </span><span class="ff4">·<span class="_ _8"> </span></span>3<span class="_ _8"> </span><span class="ff4">·<span class="_ _8"> </span></span>5<span class="_ _8"> </span><span class="ff4">·<span class="_ _8"></span></span>5<span class="_ _8"> </span><span class="ff4">·<span class="_ _8"></span></span>7<span class="_ _6"> </span><span class="ff4 ls4">···</span></div><div class="t m0 x80 h3 y432 ff2 fs0 fc0 sc0 ls0 ws0">Brounc<span class="_ _5"></span>ker’s<span class="_ _8"> </span>contin<span class="_ _5"></span>ued<span class="_ _8"> </span>fraction<span class="_ _7"> </span>expansion:</div><div class="t m0 xc8 h5 y19 ff6 fs1 fc0 sc0 ls0 ws0">π</div><div class="t m0 xc8 h5 y433 ff5 fs1 fc0 sc0 ls0 ws0">4</div><div class="t m0 x146 h3 y15 ff2 fs0 fc0 sc0 ls1 ws0">=1<span class="_ _5"></span>+</div><div class="t m0 x102 h3 y434 ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 x10b h5 y435 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x108 h3 y3e5 ff2 fs0 fc0 sc0 ls9 ws0">2+</div><div class="t m0 x95 h5 y14 ff5 fs1 fc0 sc0 ls0 ws0">3</div><div class="t m0 x124 h7 y436 ffc fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x8b h5 y2f2 ff5 fs1 fc0 sc0 ls0 ws0">2+</div><div class="t m0 x145 h7 y17 ffc fs2 fc0 sc0 ls0 ws0">5</div><div class="t m0 xd5 h7 y437 ffc fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x10b h7 y438 ffc fs2 fc0 sc0 ls0 ws0">2+</div><div class="t m0 x128 h7 y34f ffc fs2 fc0 sc0 ls0 ws0">7</div><div class="t m0 x10c h7 y242 ffc fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x145 h7 y34e ffc fs2 fc0 sc0 ls0 ws0">2+<span class="ffb">···</span></div><div class="t m0 x80 h3 y55 ff2 fs0 fc0 sc0 ls0 ws0">Gregrory’s series:</div><div class="t m0 xe1 h5 y3e8 ff6 fs1 fc0 sc0 ls0 ws0">π</div><div class="t m0 xe1 h5 y439 ff5 fs1 fc0 sc0 ls0 ws0">4</div><div class="t m0 x194 h4 y43a ff2 fs0 fc0 sc0 ls1 ws0">=1<span class="_ _5"></span><span class="ff4 ls0">−</span></div><div class="t m0 x123 h5 y3e8 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x123 h5 y439 ff5 fs1 fc0 sc0 ls0 ws0">3</div><div class="t m0 xe4 h3 y43a ff2 fs0 fc0 sc0 ls0 ws0">+</div><div class="t m0 x149 h5 y3e8 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x149 h5 y439 ff5 fs1 fc0 sc0 ls0 ws0">5</div><div class="t m0 x101 h4 y43a ff4 fs0 fc0 sc0 ls0 ws0">−</div><div class="t m0 x102 h5 y3e8 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x102 h5 y439 ff5 fs1 fc0 sc0 ls0 ws0">7</div><div class="t m0 x144 h3 y43a ff2 fs0 fc0 sc0 ls0 ws0">+</div><div class="t m0 x128 h5 y3e8 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x128 h5 y439 ff5 fs1 fc0 sc0 ls0 ws0">9</div><div class="t m0 xfd h4 y43a ff4 fs0 fc0 sc0 ls4 ws0">−<span class="_ _15"></span>···</div><div class="t m0 x80 h3 y24c ff2 fs0 fc0 sc0 ls0 ws0">Newton’s series:</div><div class="t m0 x1a4 h5 y2c8 ff6 fs1 fc0 sc0 ls0 ws0">π</div><div class="t m0 x1a9 h5 y13f ff5 fs1 fc0 sc0 ls0 ws0">6</div><div class="t m0 xc7 h3 y43b ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x86 h3 y256 ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 x86 h3 y43c ff2 fs0 fc0 sc0 ls0 ws0">2</div><div class="t m0 xfc h3 y43d ff2 fs0 fc0 sc0 ls0 ws0">+</div><div class="t m0 xe5 h3 y256 ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 x146 h4 y43c ff2 fs0 fc0 sc0 ls0 ws0">2<span class="_ _8"> </span><span class="ff4">·<span class="_ _8"> </span></span>3<span class="_ _8"> </span><span class="ff4">·<span class="_ _8"> </span></span>2</div><div class="t m0 x108 h5 y43e ff5 fs1 fc0 sc0 ls0 ws0">3</div><div class="t m0 x116 h3 y43b ff2 fs0 fc0 sc0 ls0 ws0">+</div><div class="t m0 x95 h4 y256 ff2 fs0 fc0 sc0 ls0 ws0">1<span class="_ _8"> </span><span class="ff4">·<span class="_ _8"> </span></span>3</div><div class="t m0 x8b h4 y43c ff2 fs0 fc0 sc0 ls0 ws0">2<span class="_ _8"> </span><span class="ff4">·<span class="_ _8"> </span></span>4<span class="_ _8"> </span><span class="ff4">·<span class="_ _8"> </span></span>5<span class="_ _8"> </span><span class="ff4">·<span class="_ _8"> </span></span>2</div><div class="t m0 x104 h5 y43e ff5 fs1 fc0 sc0 ls0 ws0">5</div><div class="t m0 x129 h4 y43b ff2 fs0 fc0 sc0 ls0 ws0">+<span class="_ _8"> </span><span class="ff4 ls4">···</span></div><div class="t m0 x80 h3 y32 ff2 fs0 fc0 sc0 ls0 ws0">Sharp’s series:</div><div class="t m0 x3 h5 y341 ff6 fs1 fc0 sc0 ls0 ws0">π</div><div class="t m0 x3 h5 y61 ff5 fs1 fc0 sc0 ls0 ws0">6</div><div class="t m0 x1a4 h3 y25f ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x9a h3 y2ca ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 x11e h4 y25f ff4 fs0 fc0 sc0 ls0 ws0">√</div><div class="t m0 x18d h3 y15a ff2 fs0 fc0 sc0 ls0 ws0">3</div><div class="t m0 x87 h6 y25b ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xc8 h4 y25f ff2 fs0 fc0 sc0 ls0 ws0">1<span class="_ _2"> </span><span class="ff4">−</span></div><div class="t m0 x123 h3 y2ca ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 x143 h3 y408 ff2 fs0 fc0 sc0 ls0 ws0">3</div><div class="t m0 x107 h5 y43f ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 xf2 h4 y408 ff4 fs0 fc0 sc0 ls0 ws0">·<span class="_ _8"> </span><span class="ff2">3</span></div><div class="t m0 x148 h3 y25f ff2 fs0 fc0 sc0 ls0 ws0">+</div><div class="t m0 x10a h3 y2ca ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 x101 h3 y408 ff2 fs0 fc0 sc0 ls0 ws0">3</div><div class="t m0 x8b h5 y43f ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x1a6 h4 y408 ff4 fs0 fc0 sc0 ls0 ws0">·<span class="_ _8"> </span><span class="ff2">5</span></div><div class="t m0 x124 h4 y25f ff4 fs0 fc0 sc0 ls0 ws0">−</div><div class="t m0 x96 h3 y2ca ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 x10c h3 y408 ff2 fs0 fc0 sc0 ls0 ws0">3</div><div class="t m0 xd6 h5 y43f ff5 fs1 fc0 sc0 ls0 ws0">3</div><div class="t m0 x83 h4 y408 ff4 fs0 fc0 sc0 ls0 ws0">·<span class="_ _8"> </span><span class="ff2">7</span></div><div class="t m0 x1aa h4 y25f ff2 fs0 fc0 sc0 ls0 ws0">+<span class="_ _2"> </span><span class="ff4 ls4">···</span></div><div class="t m0 xc3 h6 y25b ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x80 h3 y262 ff2 fs0 fc0 sc0 ls0 ws0">Euler’s series:</div><div class="t m0 x98 h5 y440 ff6 fs1 fc0 sc0 ls0 ws0">π</div><div class="t m0 x11e h7 y441 ffc fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x1ab h5 y15c ff5 fs1 fc0 sc0 ls0 ws0">6</div><div class="t m0 xe1 h3 y442 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x18c h5 y440 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 xc8 h5 y15c ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x88 h7 y443 ffc fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x11f h3 y442 ff2 fs0 fc0 sc0 ls0 ws0">+</div><div class="t m0 xe5 h5 y440 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x107 h5 y15c ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x120 h7 y443 ffc fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x109 h3 y442 ff2 fs0 fc0 sc0 ls0 ws0">+</div><div class="t m0 x142 h5 y440 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x149 h5 y15c ff5 fs1 fc0 sc0 ls0 ws0">3</div><div class="t m0 x6 h7 y443 ffc fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x13e h3 y442 ff2 fs0 fc0 sc0 ls0 ws0">+</div><div class="t m0 x1ac h5 y440 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x10b h5 y15c ff5 fs1 fc0 sc0 ls0 ws0">4</div><div class="t m0 x95 h7 y443 ffc fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x1a0 h3 y442 ff2 fs0 fc0 sc0 ls0 ws0">+</div><div class="t m0 x132 h5 y440 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 xd6 h5 y15c ff5 fs1 fc0 sc0 ls0 ws0">5</div><div class="t m0 x96 h7 y443 ffc fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x125 h4 y442 ff2 fs0 fc0 sc0 ls0 ws0">+<span class="_ _8"> </span><span class="ff4 ls4">···</span></div><div class="t m0 x98 h5 y444 ff6 fs1 fc0 sc0 ls0 ws0">π</div><div class="t m0 x11e h7 y4a ffc fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x1ab h5 y1ab ff5 fs1 fc0 sc0 ls0 ws0">8</div><div class="t m0 xe1 h3 y445 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x18c h5 y444 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 xc8 h5 y1ab ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x88 h7 y446 ffc fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x11f h3 y445 ff2 fs0 fc0 sc0 ls0 ws0">+</div><div class="t m0 xe5 h5 y444 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x107 h5 y1ab ff5 fs1 fc0 sc0 ls0 ws0">3</div><div class="t m0 x120 h7 y446 ffc fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x109 h3 y445 ff2 fs0 fc0 sc0 ls0 ws0">+</div><div class="t m0 x142 h5 y444 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x149 h5 y1ab ff5 fs1 fc0 sc0 ls0 ws0">5</div><div class="t m0 x6 h7 y446 ffc fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x13e h3 y445 ff2 fs0 fc0 sc0 ls0 ws0">+</div><div class="t m0 x1ac h5 y444 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x10b h5 y1ab ff5 fs1 fc0 sc0 ls0 ws0">7</div><div class="t m0 x95 h7 y446 ffc fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x1a0 h3 y445 ff2 fs0 fc0 sc0 ls0 ws0">+</div><div class="t m0 x132 h5 y444 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 xd6 h5 y1ab ff5 fs1 fc0 sc0 ls0 ws0">9</div><div class="t m0 x96 h7 y446 ffc fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x125 h4 y445 ff2 fs0 fc0 sc0 ls0 ws0">+<span class="_ _8"> </span><span class="ff4 ls4">···</span></div><div class="t m0 x98 h5 y447 ff6 fs1 fc0 sc0 ls0 ws0">π</div><div class="t m0 x11e h7 y267 ffc fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x98 h5 y448 ff5 fs1 fc0 sc0 ls0 ws0">12</div><div class="t m0 xe1 h3 y6e ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x18c h5 y447 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 xc8 h5 y448 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x88 h7 y449 ffc fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x11f h4 y6e ff4 fs0 fc0 sc0 ls0 ws0">−</div><div class="t m0 xe5 h5 y447 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x107 h5 y448 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x120 h7 y449 ffc fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x109 h3 y6e ff2 fs0 fc0 sc0 ls0 ws0">+</div><div class="t m0 x142 h5 y447 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x149 h5 y448 ff5 fs1 fc0 sc0 ls0 ws0">3</div><div class="t m0 x6 h7 y449 ffc fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x13e h4 y6e ff4 fs0 fc0 sc0 ls0 ws0">−</div><div class="t m0 x1ac h5 y447 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x10b h5 y448 ff5 fs1 fc0 sc0 ls0 ws0">4</div><div class="t m0 x95 h7 y449 ffc fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x1a0 h3 y6e ff2 fs0 fc0 sc0 ls0 ws0">+</div><div class="t m0 x132 h5 y447 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 xd6 h5 y448 ff5 fs1 fc0 sc0 ls0 ws0">5</div><div class="t m0 x96 h7 y449 ffc fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x125 h4 y6e ff4 fs0 fc0 sc0 ls4 ws0">−<span class="_ _15"></span>···</div><div class="t m0 x8 h3 y34d ff2 fs0 fc0 sc0 ls0 ws0">Deriv<span class="_ _5"></span>ativ<span class="_ _5"></span>es:</div><div class="t m0 x8 h2 y44a ff1 fs0 fc0 sc0 ls0 ws0">1.</div><div class="t m0 xe9 h3 y2f5 ff3 fs0 fc0 sc0 ls0 ws0">d<span class="ff2">(</span>cu<span class="ff2">)</span></div><div class="t m0 x11b h3 y11c ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x71 h3 y44b ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff3">c</span></div><div class="t m0 x18a h3 y2f5 ff3 fs0 fc0 sc0 ls0 ws0">du</div><div class="t m0 x18a h3 y11c ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x9d h3 y44b ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _2a"> </span><span class="ff1">2.</span></div><div class="t m0 xa8 h3 y2f5 ff3 fs0 fc0 sc0 ls0 ws0">d<span class="ff2">(</span>u<span class="_ _8"> </span><span class="ff2">+<span class="_ _8"> </span></span>v<span class="_ _3"></span><span class="ff2">)</span></div><div class="t m0 x197 h3 y11c ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x2f h3 y44b ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x76 h3 y2f5 ff3 fs0 fc0 sc0 ls0 ws0">du</div><div class="t m0 x76 h3 y11c ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x9f h3 y44b ff2 fs0 fc0 sc0 ls0 ws0">+</div><div class="t m0 x17b h3 y2f5 ff3 fs0 fc0 sc0 ls0 ws0">dv</div><div class="t m0 x17b h3 y11c ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x43 h3 y44b ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _2a"> </span><span class="ff1">3.</span></div><div class="t m0 x19 h3 y2f5 ff3 fs0 fc0 sc0 ls0 ws0">d<span class="ff2">(</span>uv<span class="_ _3"></span><span class="ff2">)</span></div><div class="t m0 x52 h3 y11c ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x53 h3 y44b ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff3">u</span></div><div class="t m0 x65 h3 y2f5 ff3 fs0 fc0 sc0 ls0 ws0">dv</div><div class="t m0 x65 h3 y11c ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 xe0 h3 y44b ff2 fs0 fc0 sc0 ls0 ws0">+<span class="_ _8"> </span><span class="ff3">v</span></div><div class="t m0 xd3 h3 y2f5 ff3 fs0 fc0 sc0 ls0 ws0">du</div><div class="t m0 xd3 h3 y11c ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 xa7 h3 y44b ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x8 h2 y19 ff1 fs0 fc0 sc0 ls0 ws0">4.</div><div class="t m0 xe9 h3 y44c ff3 fs0 fc0 sc0 ls0 ws0">d<span class="ff2">(</span>u</div><div class="t m0 x153 h5 y44d ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x26 h3 y44e ff2 fs0 fc0 sc0 ls0 ws0">)</div><div class="t m0 x151 h3 y44f ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x177 h3 y19 ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff3">nu</span></div><div class="t m0 x29 h5 y11 ff6 fs1 fc0 sc0 ls0 ws0">n<span class="ff8">−<span class="ff5">1</span></span></div><div class="t m0 x49 h3 y44e ff3 fs0 fc0 sc0 ls0 ws0">du</div><div class="t m0 x49 h3 y44f ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x68 h3 y19 ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _6b"> </span><span class="ff1">5.</span></div><div class="t m0 x3d h3 y44c ff3 fs0 fc0 sc0 ls0 ws0">d<span class="ff2">(</span>u/v<span class="_ _3"></span><span class="ff2">)</span></div><div class="t m0 x127 h3 y44f ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x3f h3 y19 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x41 h3 y51 ff3 fs0 fc0 sc0 ls0 ws0">v</div><div class="t m0 x122 h6 y11a ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x9f h5 y450 ff6 fs1 fc0 sc0 ls0 ws0">du</div><div class="t m0 x9f h5 y451 ff6 fs1 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x111 h6 y11a ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x163 h4 y51 ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff3">u</span></div><div class="t m0 xec h6 y11a ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xfb h5 y450 ff6 fs1 fc0 sc0 ls0 ws0">dv</div><div class="t m0 xfb h5 y451 ff6 fs1 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x61 h6 y11a ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x162 h3 y44f ff3 fs0 fc0 sc0 ls0 ws0">v</div><div class="t m0 x186 h5 y15 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x18 h3 y19 ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _6b"> </span><span class="ff1">6.</span></div><div class="t m0 x64 h3 y44e ff3 fs0 fc0 sc0 ls0 ws0">d<span class="ff2">(</span>e</div><div class="t m0 x168 h5 y44d ff6 fs1 fc0 sc0 ls0 ws0">cu</div><div class="t m0 x20 h3 y44e ff2 fs0 fc0 sc0 ls0 ws0">)</div><div class="t m0 x166 h3 y44f ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 xbd h3 y19 ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff3">ce</span></div><div class="t m0 xba h5 y11 ff6 fs1 fc0 sc0 ls0 ws0">cu</div><div class="t m0 xd3 h3 y44e ff3 fs0 fc0 sc0 ls0 ws0">du</div><div class="t m0 xd3 h3 y44f ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 xa7 h3 y19 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x8 h2 y21 ff1 fs0 fc0 sc0 ls0 ws0">7.</div><div class="t m0 xe9 h3 y54 ff3 fs0 fc0 sc0 ls0 ws0">d<span class="ff2">(</span>c</div><div class="t m0 xfa h5 y452 ff6 fs1 fc0 sc0 ls0 ws0">u</div><div class="t m0 x25 h3 y54 ff2 fs0 fc0 sc0 ls0 ws0">)</div><div class="t m0 x11b h3 y453 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 xb h3 y21 ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span>(ln<span class="_ _6"> </span><span class="ff3">c</span>)<span class="ff3">c</span></div><div class="t m0 x47 h5 y454 ff6 fs1 fc0 sc0 ls0 ws0">u</div><div class="t m0 x5b h3 y54 ff3 fs0 fc0 sc0 ls0 ws0">du</div><div class="t m0 x5b h3 y453 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x48 h3 y21 ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _6c"> </span><span class="ff1">8.</span></div><div class="t m0 x166 h3 y54 ff3 fs0 fc0 sc0 ls0 ws0">d<span class="ff2">(ln<span class="_ _6"> </span></span>u<span class="ff2">)</span></div><div class="t m0 x199 h3 y453 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 xa6 h3 y21 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 xd2 h3 y54 ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 xd2 h3 y453 ff3 fs0 fc0 sc0 ls0 ws0">u</div><div class="t m0 xd3 h3 y54 ff3 fs0 fc0 sc0 ls0 ws0">du</div><div class="t m0 xd3 h3 y453 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 xa7 h3 y21 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x8 h2 y3ff ff1 fs0 fc0 sc0 ls0 ws0">9.</div><div class="t m0 xe9 h3 y131 ff3 fs0 fc0 sc0 ls0 ws0">d<span class="ff2">(sin<span class="_ _6"> </span></span>u<span class="ff2">)</span></div><div class="t m0 xee h3 y2c6 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x28 h3 y3ff ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span>cos<span class="_ _6"> </span><span class="ff3">u</span></div><div class="t m0 x14e h3 y131 ff3 fs0 fc0 sc0 ls0 ws0">du</div><div class="t m0 x14e h3 y2c6 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x48 h3 y3ff ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _6d"> </span><span class="ff1">10.</span></div><div class="t m0 x36 h3 y131 ff3 fs0 fc0 sc0 ls0 ws0">d<span class="ff2">(cos<span class="_ _6"> </span></span>u<span class="ff2">)</span></div><div class="t m0 xbb h3 y2c6 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 xdf h4 y3ff ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff4">−<span class="_ _6"></span></span>sin<span class="_ _8"> </span><span class="ff3">u</span></div><div class="t m0 xd3 h3 y131 ff3 fs0 fc0 sc0 ls0 ws0">du</div><div class="t m0 xd3 h3 y2c6 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 xa7 h3 y3ff ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x8 h2 y254 ff1 fs0 fc0 sc0 ls0 ws0">11.</div><div class="t m0 x11b h3 y13c ff3 fs0 fc0 sc0 ls0 ws0">d<span class="ff2">(tan<span class="_ _6"> </span></span>u<span class="ff2">)</span></div><div class="t m0 x25 h3 y3b4 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 xef h3 y254 ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span>sec</div><div class="t m0 x49 h5 y156 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x156 h3 y254 ff3 fs0 fc0 sc0 ls0 ws0">u</div><div class="t m0 x48 h3 y13c ff3 fs0 fc0 sc0 ls0 ws0">du</div><div class="t m0 x48 h3 y3b4 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 xab h3 y254 ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _6e"> </span><span class="ff1">12.</span></div><div class="t m0 x17e h3 y13c ff3 fs0 fc0 sc0 ls0 ws0">d<span class="ff2">(cot<span class="_ _6"> </span></span>u<span class="ff2">)</span></div><div class="t m0 x1d h3 y3b4 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x20 h3 y254 ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span>csc</div><div class="t m0 x182 h5 y156 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x105 h3 y254 ff3 fs0 fc0 sc0 ls0 ws0">u</div><div class="t m0 xd3 h3 y13c ff3 fs0 fc0 sc0 ls0 ws0">du</div><div class="t m0 xd3 h3 y3b4 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 xa7 h3 y254 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x8 h2 y455 ff1 fs0 fc0 sc0 ls0 ws0">13.</div><div class="t m0 x11b h3 y142 ff3 fs0 fc0 sc0 ls0 ws0">d<span class="ff2">(sec<span class="_ _6"> </span></span>u<span class="ff2">)</span></div><div class="t m0 x1ad h3 y456 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x18a h3 y457 ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span>tan<span class="_ _6"> </span><span class="ff3">u </span>sec<span class="_ _8"> </span><span class="ff3">u</span></div><div class="t m0 xac h3 y142 ff3 fs0 fc0 sc0 ls0 ws0">du</div><div class="t m0 xac h3 y456 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x11 h3 y457 ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _6f"> </span><span class="ff1">14.</span></div><div class="t m0 x60 h3 y142 ff3 fs0 fc0 sc0 ls0 ws0">d<span class="ff2">(csc<span class="_ _6"> </span></span>u<span class="ff2">)</span></div><div class="t m0 x4f h3 y456 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x1c h4 y457 ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff4">−<span class="_ _6"></span></span>cot<span class="_ _8"> </span><span class="ff3">u<span class="_ _7"> </span></span>csc<span class="_ _8"> </span><span class="ff3">u</span></div><div class="t m0 xd3 h3 y142 ff3 fs0 fc0 sc0 ls0 ws0">du</div><div class="t m0 xd3 h3 y456 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 xa7 h3 y457 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x8 h2 y458 ff1 fs0 fc0 sc0 ls0 ws0">15.</div><div class="t m0 x11b h3 y459 ff3 fs0 fc0 sc0 ls0 ws0">d<span class="ff2">(arcsin<span class="_ _6"> </span></span>u<span class="ff2">)</span></div><div class="t m0 x56 h3 y63 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x9d h3 y45a ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x178 h3 y45b ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 x49 h4 y45c ff4 fs0 fc0 sc0 ls0 ws0">√</div><div class="t m0 xe h4 y45d ff2 fs0 fc0 sc0 ls0 ws0">1<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span><span class="ff3">u</span></span></div><div class="t m0 x170 h5 y61 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 xa8 h3 y45b ff3 fs0 fc0 sc0 ls0 ws0">du</div><div class="t m0 xa8 h3 y63 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x197 h3 y45a ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _1c"> </span><span class="ff1">16.</span></div><div class="t m0 x61 h3 y45b ff3 fs0 fc0 sc0 ls0 ws0">d<span class="ff2">(arccos<span class="_ _6"> </span></span>u<span class="ff2">)</span></div><div class="t m0 x6f h3 y63 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 xd0 h3 y45a ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 xa4 h4 y45b ff4 fs0 fc0 sc0 ls0 ws0">−<span class="ff2">1</span></div><div class="t m0 xf0 h4 y45c ff4 fs0 fc0 sc0 ls0 ws0">√</div><div class="t m0 xbd h4 y45d ff2 fs0 fc0 sc0 ls0 ws0">1<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span><span class="ff3">u</span></span></div><div class="t m0 x105 h5 y61 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 xd3 h3 y45b ff3 fs0 fc0 sc0 ls0 ws0">du</div><div class="t m0 xd3 h3 y63 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 xa7 h3 y45a ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x8 h2 y285 ff1 fs0 fc0 sc0 ls0 ws0">17.</div><div class="t m0 x11b h3 y45e ff3 fs0 fc0 sc0 ls0 ws0">d<span class="ff2">(arctan<span class="_ _6"> </span></span>u<span class="ff2">)</span></div><div class="t m0 x154 h3 y45f ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 xb3 h3 y285 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x48 h3 y460 ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 x5b h3 y45f ff2 fs0 fc0 sc0 ls9 ws0">1+<span class="ff3 ls0">u</span></div><div class="t m0 xab h5 y461 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 xac h3 y460 ff3 fs0 fc0 sc0 ls0 ws0">du</div><div class="t m0 xac h3 y45f ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x11 h3 y285 ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _70"> </span><span class="ff1">18.</span></div><div class="t m0 x4f h3 y460 ff3 fs0 fc0 sc0 ls0 ws0">d<span class="ff2">(arccot<span class="_ _6"> </span></span>u<span class="ff2">)</span></div><div class="t m0 x51 h3 y45f ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 xdf h3 y285 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x23 h4 y460 ff4 fs0 fc0 sc0 ls0 ws0">−<span class="ff2">1</span></div><div class="t m0 xbd h3 y45f ff2 fs0 fc0 sc0 ls2 ws0">1+<span class="ff3 ls0">u</span></div><div class="t m0 x105 h5 y461 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 xd3 h3 y460 ff3 fs0 fc0 sc0 ls0 ws0">du</div><div class="t m0 xd3 h3 y45f ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 xa7 h3 y285 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x8 h2 y1a5 ff1 fs0 fc0 sc0 ls0 ws0">19.</div><div class="t m0 x11b h3 y288 ff3 fs0 fc0 sc0 ls0 ws0">d<span class="ff2">(arcsec<span class="_ _6"> </span></span>u<span class="ff2">)</span></div><div class="t m0 x154 h3 y45 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x9d h3 y1a5 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x15f h3 y288 ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 x49 h3 y1ad ff3 fs0 fc0 sc0 ls0 ws0">u</div><div class="t m0 x156 h4 y1a5 ff4 fs0 fc0 sc0 ls0 ws0">√</div><div class="t m0 x48 h4 y1ad ff2 fs0 fc0 sc0 ls0 ws0">1<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span><span class="ff3">u</span></span></div><div class="t m0 xa8 h5 y269 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x11 h3 y288 ff3 fs0 fc0 sc0 ls0 ws0">du</div><div class="t m0 x11 h3 y45 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x127 h3 y1a5 ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _71"> </span><span class="ff1">20.</span></div><div class="t m0 x7a h3 y288 ff3 fs0 fc0 sc0 ls0 ws0">d<span class="ff2">(arccsc<span class="_ _6"> </span></span>u<span class="ff2">)</span></div><div class="t m0 x113 h3 y45 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 xcf h3 y1a5 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 xbd h4 y288 ff4 fs0 fc0 sc0 ls0 ws0">−<span class="ff2">1</span></div><div class="t m0 x199 h3 y1ad ff3 fs0 fc0 sc0 ls0 ws0">u</div><div class="t m0 xf0 h4 y1a5 ff4 fs0 fc0 sc0 ls0 ws0">√</div><div class="t m0 xbd h4 y1ad ff2 fs0 fc0 sc0 ls0 ws0">1<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span><span class="ff3">u</span></span></div><div class="t m0 x105 h5 y269 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 xd3 h3 y288 ff3 fs0 fc0 sc0 ls0 ws0">du</div><div class="t m0 xd3 h3 y45 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 xa7 h3 y1a5 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x8 h2 y462 ff1 fs0 fc0 sc0 ls0 ws0">21.</div><div class="t m0 x11b h3 y161 ff3 fs0 fc0 sc0 ls0 ws0">d<span class="ff2">(sinh<span class="_ _6"> </span></span>u<span class="ff2">)</span></div><div class="t m0 x57 h3 y463 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x46 h3 y464 ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span>cosh<span class="_ _6"> </span><span class="ff3">u</span></div><div class="t m0 x15f h3 y161 ff3 fs0 fc0 sc0 ls0 ws0">du</div><div class="t m0 x15f h3 y463 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x170 h3 y464 ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _30"> </span><span class="ff1">22.</span></div><div class="t m0 x1b h3 y161 ff3 fs0 fc0 sc0 ls0 ws0">d<span class="ff2">(cosh<span class="_ _6"> </span></span>u<span class="ff2">)</span></div><div class="t m0 x1c h3 y463 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x20 h3 y464 ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span>sinh<span class="_ _6"> </span><span class="ff3">u</span></div><div class="t m0 xd3 h3 y161 ff3 fs0 fc0 sc0 ls0 ws0">du</div><div class="t m0 xd3 h3 y463 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 xa7 h3 y464 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x8 h2 y465 ff1 fs0 fc0 sc0 ls0 ws0">23.</div><div class="t m0 x11b h3 y368 ff3 fs0 fc0 sc0 ls0 ws0">d<span class="ff2">(tanh<span class="_ _6"> </span></span>u<span class="ff2">)</span></div><div class="t m0 x26 h3 y466 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x29 h3 y467 ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span>sec<span class="_ _5"></span>h</div><div class="t m0 x68 h5 y468 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x9e h3 y467 ff3 fs0 fc0 sc0 ls0 ws0">u</div><div class="t m0 xab h3 y368 ff3 fs0 fc0 sc0 ls0 ws0">du</div><div class="t m0 xab h3 y466 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x12b h3 y467 ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _2d"> </span><span class="ff1">24.</span></div><div class="t m0 x50 h3 y368 ff3 fs0 fc0 sc0 ls0 ws0">d<span class="ff2">(coth<span class="_ _6"> </span></span>u<span class="ff2">)</span></div><div class="t m0 x1b h3 y466 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 xbc h4 y467 ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff4">−<span class="_ _6"></span></span>csch</div><div class="t m0 x182 h5 y468 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x105 h3 y467 ff3 fs0 fc0 sc0 ls0 ws0">u</div><div class="t m0 xd3 h3 y368 ff3 fs0 fc0 sc0 ls0 ws0">du</div><div class="t m0 xd3 h3 y466 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 xa7 h3 y467 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x8 h2 y28e ff1 fs0 fc0 sc0 ls0 ws0">25.</div><div class="t m0 x11b h3 y3f1 ff3 fs0 fc0 sc0 ls0 ws0">d<span class="ff2">(sec<span class="_ _5"></span>h<span class="_ _8"> </span><span class="ff3">u<span class="_ _5"></span><span class="ff2">)</span></span></span></div><div class="t m0 x57 h3 y28d ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x46 h4 y28e ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff4">−<span class="_ _6"></span></span>sech<span class="_ _6"> </span><span class="ff3">u </span>tanh<span class="_ _8"> </span><span class="ff3">u</span></div><div class="t m0 x2e h3 y3f1 ff3 fs0 fc0 sc0 ls0 ws0">du</div><div class="t m0 x2e h3 y28d ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x198 h3 y28e ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _5e"> </span><span class="ff1">26.</span></div><div class="t m0 x11d h3 y3f1 ff3 fs0 fc0 sc0 ls0 ws0">d<span class="ff2">(csc<span class="_ _5"></span>h<span class="_ _8"> </span><span class="ff3">u<span class="_ _5"></span><span class="ff2">)</span></span></span></div><div class="t m0 xa1 h3 y28d ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x19 h4 y28e ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff4">−<span class="_ _6"></span></span>csch<span class="_ _6"> </span><span class="ff3">u </span>coth<span class="_ _8"> </span><span class="ff3">u</span></div><div class="t m0 xd3 h3 y3f1 ff3 fs0 fc0 sc0 ls0 ws0">du</div><div class="t m0 xd3 h3 y28d ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 xa7 h3 y28e ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x8 h2 y270 ff1 fs0 fc0 sc0 ls0 ws0">27.</div><div class="t m0 x11b h3 y469 ff3 fs0 fc0 sc0 ls0 ws0">d<span class="ff2">(arcsinh<span class="_ _6"> </span></span>u<span class="ff2">)</span></div><div class="t m0 xb h3 y1ff ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x2b h3 y270 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x5c h3 y46a ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 x2a h4 y270 ff4 fs0 fc0 sc0 ls0 ws0">√</div><div class="t m0 x48 h3 y1c2 ff2 fs0 fc0 sc0 ls2 ws0">1+<span class="ff3 ls0">u</span></div><div class="t m0 x12b h5 yb0 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x15a h3 y46a ff3 fs0 fc0 sc0 ls0 ws0">du</div><div class="t m0 x15a h3 y1ff ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x127 h3 y270 ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _72"> </span><span class="ff1">28.</span></div><div class="t m0 x7a h3 y46a ff3 fs0 fc0 sc0 ls0 ws0">d<span class="ff2">(arccosh<span class="_ _6"> </span></span>u<span class="ff2">)</span></div><div class="t m0 x62 h3 y1ff ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 xd0 h3 y270 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x23 h3 y46a ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 xf0 h4 y270 ff4 fs0 fc0 sc0 ls0 ws0">√</div><div class="t m0 xbd h3 y1c2 ff3 fs0 fc0 sc0 ls0 ws0">u</div><div class="t m0 xd1 h5 yb0 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x7d h4 y1c2 ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff2">1</span></div><div class="t m0 xd3 h3 y46a ff3 fs0 fc0 sc0 ls0 ws0">du</div><div class="t m0 xd3 h3 y1ff ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 xa7 h3 y270 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x8 h2 y418 ff1 fs0 fc0 sc0 ls0 ws0">29.</div><div class="t m0 x11b h3 y374 ff3 fs0 fc0 sc0 ls0 ws0">d<span class="ff2">(arctanh<span class="_ _6"> </span></span>u<span class="ff2">)</span></div><div class="t m0 x71 h3 y375 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x47 h3 y418 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x15f h3 y374 ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 xe h4 y375 ff2 fs0 fc0 sc0 ls0 ws0">1<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span><span class="ff3">u</span></span></div><div class="t m0 x170 h5 y41a ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 xa8 h3 y374 ff3 fs0 fc0 sc0 ls0 ws0">du</div><div class="t m0 xa8 h3 y375 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x10 h3 y418 ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _6f"> </span><span class="ff1">30.</span></div><div class="t m0 x35 h3 y374 ff3 fs0 fc0 sc0 ls0 ws0">d<span class="ff2">(arccoth<span class="_ _6"> </span></span>u<span class="ff2">)</span></div><div class="t m0 x19 h3 y375 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 xdf h3 y418 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x7d h3 y374 ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 xbd h3 y375 ff3 fs0 fc0 sc0 ls0 ws0">u</div><div class="t m0 xd1 h5 y41a ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x7d h4 y375 ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff2">1</span></div><div class="t m0 xd3 h3 y374 ff3 fs0 fc0 sc0 ls0 ws0">du</div><div class="t m0 xd3 h3 y375 ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 xa7 h3 y418 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x8 h2 y46b ff1 fs0 fc0 sc0 ls0 ws0">31.</div><div class="t m0 x11b h3 y46c ff3 fs0 fc0 sc0 ls0 ws0">d<span class="ff2">(arcsec<span class="_ _5"></span>h<span class="_ _8"> </span><span class="ff3">u<span class="_ _5"></span><span class="ff2">)</span></span></span></div><div class="t m0 xb h3 y46d ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x2b h3 y46e ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x5c h4 y46f ff4 fs0 fc0 sc0 ls0 ws0">−<span class="ff2">1</span></div><div class="t m0 x2a h3 y470 ff3 fs0 fc0 sc0 ls0 ws0">u</div><div class="t m0 x68 h4 y471 ff4 fs0 fc0 sc0 ls0 ws0">√</div><div class="t m0 x175 h4 y470 ff2 fs0 fc0 sc0 ls0 ws0">1<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span><span class="ff3">u</span></span></div><div class="t m0 x75 h5 yb7 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x10 h3 y46f ff3 fs0 fc0 sc0 ls0 ws0">du</div><div class="t m0 x10 h3 y46d ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x139 h3 y46e ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _41"> </span><span class="ff1">32.</span></div><div class="t m0 xec h3 y46f ff3 fs0 fc0 sc0 ls0 ws0">d<span class="ff2">(arccsc<span class="_ _5"></span>h<span class="_ _8"> </span><span class="ff3">u<span class="_ _5"></span><span class="ff2">)</span></span></span></div><div class="t m0 x18 h3 y46d ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x1d h3 y46e ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x21 h4 y46f ff4 fs0 fc0 sc0 ls0 ws0">−<span class="ff2">1</span></div><div class="t m0 xd0 h4 y470 ff4 fs0 fc0 sc0 ls0 ws0">|<span class="ff3">u</span>|</div><div class="t m0 xf0 h4 y471 ff4 fs0 fc0 sc0 ls0 ws0">√</div><div class="t m0 xbd h3 y470 ff2 fs0 fc0 sc0 ls2 ws0">1+<span class="ff3 ls0">u</span></div><div class="t m0 x105 h5 yb7 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 xd3 h3 y46f ff3 fs0 fc0 sc0 ls0 ws0">du</div><div class="t m0 xd3 h3 y46d ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 xa7 h3 y46e ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x8 h3 y183 ff2 fs0 fc0 sc0 ls0 ws0">In<span class="_ _5"></span>tegrals:</div><div class="t m0 x8 h2 y18c ff1 fs0 fc0 sc0 ls0 ws0">1.</div><div class="t m0 x157 h6 y378 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xee h3 y18c ff3 fs0 fc0 sc0 ls0 ws0">cu<span class="_ _6"> </span>dx<span class="_ _7"> </span><span class="ff2">=<span class="_ _7"> </span></span>c</div><div class="t m0 x73 h6 y378 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x47 h3 y18c ff3 fs0 fc0 sc0 ls0 ws0">u<span class="_ _6"> </span>dx,<span class="_ _73"> </span><span class="ff1">2.</span></div><div class="t m0 x34 h6 y378 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xec h3 y18c ff2 fs0 fc0 sc0 ls0 ws0">(<span class="ff3">u<span class="_ _8"> </span></span>+<span class="_ _8"> </span><span class="ff3">v<span class="_ _3"></span></span>)<span class="_ _6"> </span><span class="ff3">dx<span class="_ _7"> </span></span>=</div><div class="t m0 xb9 h6 y378 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x53 h3 y18c ff3 fs0 fc0 sc0 ls4 ws0">ud<span class="_ _1f"></span>x<span class="_ _3"></span><span class="ff2 ls0">+</span></div><div class="t m0 x7d h6 y378 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x165 h3 y18c ff3 fs0 fc0 sc0 ls0 ws0">v<span class="_ _8"> </span>dx,</div><div class="t m0 x8 h2 yce ff1 fs0 fc0 sc0 ls0 ws0">3.</div><div class="t m0 x157 h6 ybd ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xee h3 yce ff3 fs0 fc0 sc0 ls0 ws0">x</div><div class="t m0 x19e h5 y397 ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x56 h3 yce ff3 fs0 fc0 sc0 ls0 ws0">dx<span class="_ _7"> </span><span class="ff2">=</span></div><div class="t m0 xb3 h3 yc0 ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 x137 h3 y472 ff3 fs0 fc0 sc0 ls0 ws0">n<span class="_ _8"> </span><span class="ff2 ls2">+1</span></div><div class="t m0 x2a h3 yce ff3 fs0 fc0 sc0 ls0 ws0">x</div><div class="t m0 x68 h5 y397 ff6 fs1 fc0 sc0 ls0 ws0">n<span class="ff5">+1</span></div><div class="t m0 x2d h4 yce ff3 fs0 fc0 sc0 ls3 ws0">,n<span class="_ _a"></span><span class="ff4 ls0"><span class="ff2">=<span class="_ _7"> </span></span>−<span class="ff2">1<span class="ff3">,<span class="_ _74"> </span><span class="ff1">4.</span></span></span></span></div><div class="t m0 xa0 h6 ybd ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x181 h3 yc0 ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 x181 h3 y472 ff3 fs0 fc0 sc0 ls0 ws0">x</div><div class="t m0 x17d h3 yce ff3 fs0 fc0 sc0 ls0 ws0">dx<span class="_ _7"> </span><span class="ff2 ls1">=l<span class="_ _b"></span>n<span class="_ _3d"></span><span class="ff3 ls0">x,<span class="_ _74"> </span><span class="ff1">5.</span></span></span></div><div class="t m0 x37 h6 ybd ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x21 h3 yce ff3 fs0 fc0 sc0 ls0 ws0">e</div><div class="t m0 x7e h5 y397 ff6 fs1 fc0 sc0 ls0 ws0">x</div><div class="t m0 x23 h3 yce ff3 fs0 fc0 sc0 ls0 ws0">dx<span class="_ _7"> </span><span class="ff2">=<span class="_ _7"> </span></span>e</div><div class="t m0 x185 h5 y397 ff6 fs1 fc0 sc0 ls0 ws0">x</div><div class="t m0 xa7 h3 yce ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x8 h2 y215 ff1 fs0 fc0 sc0 ls0 ws0">6.</div><div class="t m0 x157 h6 y18e ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x26 h3 y1db ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x173 h3 y39b ff2 fs0 fc0 sc0 ls9 ws0">1+<span class="ff3 ls0">x</span></div><div class="t m0 x58 h5 y473 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x18a h3 y215 ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span>arctan<span class="_ _6"> </span><span class="ff3">x,<span class="_ _75"> </span><span class="ff1">7.</span></span></div><div class="t m0 x7a h6 y18e ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x18 h3 y215 ff3 fs0 fc0 sc0 ls0 ws0">u</div><div class="t m0 x113 h3 y1db ff3 fs0 fc0 sc0 ls0 ws0">dv</div><div class="t m0 xb7 h3 y39b ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x19 h4 y215 ff3 fs0 fc0 sc0 ls0 ws0">dx<span class="_ _7"> </span><span class="ff2">=<span class="_ _7"> </span></span>uv<span class="_ _7"> </span><span class="ff4">−</span></div><div class="t m0 x7e h6 y18e ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x7d h3 y215 ff3 fs0 fc0 sc0 ls0 ws0">v</div><div class="t m0 xa5 h3 y1db ff3 fs0 fc0 sc0 ls0 ws0">du</div><div class="t m0 xa5 h3 y39b ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x38 h3 y215 ff3 fs0 fc0 sc0 ls0 ws0">dx,</div><div class="t m0 x8 h2 ydf ff1 fs0 fc0 sc0 ls0 ws0">8.</div><div class="t m0 x157 h6 y197 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xee h4 ydf ff2 fs0 fc0 sc0 ls0 ws0">sin<span class="_ _6"> </span><span class="ff3 ls4">xd<span class="_ _1f"></span>x<span class="_ _2"></span><span class="ff2 ls0">=<span class="_ _7"> </span><span class="ff4">−<span class="_ _6"></span></span>cos<span class="_ _8"> </span><span class="ff3">x,<span class="_ _76"> </span><span class="ff1">9.</span></span></span></span></div><div class="t m0 x1c h6 y197 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x1e h3 ydf ff2 fs0 fc0 sc0 ls0 ws0">cos<span class="_ _6"> </span><span class="ff3 ls4">xd<span class="_ _1f"></span>x<span class="_ _2"></span><span class="ff2 ls0">=<span class="_ _7"> </span>sin<span class="_ _6"> </span><span class="ff3">x,</span></span></span></div><div class="t m0 x8 h2 y474 ff1 fs0 fc0 sc0 ls0 ws0">10.</div><div class="t m0 x16e h6 y382 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x19e h4 y474 ff2 fs0 fc0 sc0 ls0 ws0">tan<span class="_ _6"> </span><span class="ff3 ls4">xd<span class="_ _1f"></span>x<span class="_ _2"></span><span class="ff2 ls0">=<span class="_ _7"> </span><span class="ff4">−<span class="_ _6"></span></span>ln<span class="_ _8"> </span><span class="ff4">|<span class="_ _2"></span></span>cos<span class="_ _8"> </span><span class="ff3">x<span class="_ _5"></span><span class="ff4">|<span class="ff3">,<span class="_ _77"> </span><span class="ff1">11.</span></span></span></span></span></span></div><div class="t m0 xb8 h6 y382 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x63 h4 y474 ff2 fs0 fc0 sc0 ls0 ws0">cot<span class="_ _6"> </span><span class="ff3 ls4">xd<span class="_ _1f"></span>x<span class="_ _2"></span><span class="ff2 ls1">=l<span class="_ _b"></span>n<span class="_ _3d"></span><span class="ff4 ls0">|<span class="_ _6"></span><span class="ff2">cos<span class="_ _8"> </span><span class="ff3">x<span class="_ _5"></span><span class="ff4">|<span class="ff3">,</span></span></span></span></span></span></span></div><div class="t m0 x8 h2 y475 ff1 fs0 fc0 sc0 ls0 ws0">12.</div><div class="t m0 x16e h6 y2b4 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x19e h4 y475 ff2 fs0 fc0 sc0 ls0 ws0">sec<span class="_ _6"> </span><span class="ff3 ls4">xd<span class="_ _1f"></span>x<span class="_ _2"></span><span class="ff2 ls1">=l<span class="_ _b"></span>n<span class="_ _3d"></span><span class="ff4 ls0">|<span class="_ _6"></span><span class="ff2">sec<span class="_ _8"> </span><span class="ff3">x<span class="_ _8"> </span></span>+<span class="_ _8"> </span>tan<span class="_ _2"> </span><span class="ff3">x</span></span>|<span class="ff3">,<span class="_ _3e"> </span><span class="ff1">13.</span></span></span></span></span></div><div class="t m0 x79 h6 y2b4 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xa1 h4 y475 ff2 fs0 fc0 sc0 ls0 ws0">csc<span class="_ _6"> </span><span class="ff3 ls4">xd<span class="_ _1f"></span>x<span class="_ _2"></span><span class="ff2 ls1">=l<span class="_ _b"></span>n<span class="_ _3d"></span><span class="ff4 ls0">|<span class="_ _6"></span><span class="ff2">csc<span class="_ _8"> </span><span class="ff3">x<span class="_ _8"> </span></span>+<span class="_ _8"> </span>cot<span class="_ _2"></span><span class="ff3">x</span></span>|<span class="ff3">,</span></span></span></span></div><div class="t m0 x8 h2 y476 ff1 fs0 fc0 sc0 ls0 ws0">14.</div><div class="t m0 x16e h6 yf8 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x19e h3 y476 ff2 fs0 fc0 sc0 ls0 ws0">arcsin</div><div class="t m0 x16f h5 y3a4 ff6 fs1 fc0 sc0 ls0 ws0">x</div><div class="t m0 x16f h5 y477 ff6 fs1 fc0 sc0 ls0 ws0">a</div><div class="t m0 x29 h3 y478 ff3 fs0 fc0 sc0 ls0 ws0">dx<span class="_ _7"> </span><span class="ff2">=<span class="_ _7"> </span>arcsin</span></div><div class="t m0 x10e h5 y3a4 ff6 fs1 fc0 sc0 ls0 ws0">x</div><div class="t m0 x10e h5 y477 ff6 fs1 fc0 sc0 ls0 ws0">a</div><div class="t m0 x15a h3 y478 ff2 fs0 fc0 sc0 ls0 ws0">+</div><div class="t m0 xaf h6 y105 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x14 h3 y478 ff3 fs0 fc0 sc0 ls0 ws0">a</div><div class="t m0 xcc h5 y2ef ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x6b h4 y478 ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff3">x</span></div><div class="t m0 xce h5 y2ef ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x77 h3 y478 ff3 fs0 fc0 sc0 ls3 ws0">,a<span class="_ _a"></span>&gt;<span class="_ _a"></span><span class="ff2 ls0">0<span class="ff3">,</span></span></div><div class="t m0 x18b h3 y390 ff2 fs0 fc0 sc0 ls0 ws0">P<span class="_ _5"></span>artial F<span class="_ _5"></span>ractions</div><div class="t m0 x80 h3 yaa ff2 fs0 fc0 sc0 ls0 ws0">Let<span class="_ _34"> </span><span class="ff3">N<span class="_ _2"></span></span>(<span class="ff3">x</span>)<span class="_ _0"> </span>and<span class="_ _34"> </span><span class="ff3">D<span class="_ _3"></span></span>(<span class="ff3">x</span>)<span class="_ _34"> </span>b<span class="_ _3"></span>e<span class="_ _34"> </span>p<span class="_ _3"></span>olynomial<span class="_ _34"> </span>func-</div><div class="t m0 x80 h3 yab ff2 fs0 fc0 sc0 ls0 ws0">tions<span class="_ _37"> </span>of<span class="_ _78"> </span><span class="ff3">x</span>.<span class="_ _79"> </span>W<span class="_ _5"></span>e<span class="_ _17"> </span>can<span class="_ _78"> </span>break<span class="_ _37"> </span>down</div><div class="t m0 x80 h3 y82 ff3 fs0 fc0 sc0 ls0 ws0">N<span class="_ _2"></span><span class="ff2">(</span>x<span class="ff2">)</span>/D<span class="_ _3"></span><span class="ff2">(</span>x<span class="ff2">)<span class="_ _34"> </span>using<span class="_ _0"> </span>partial<span class="_ _34"> </span>fraction<span class="_ _34"> </span>expan-</span></div><div class="t m0 x80 h3 y88 ff2 fs0 fc0 sc0 ls0 ws0">sion.<span class="_ _21"> </span>First,<span class="_ _34"> </span>if<span class="_ _34"> </span>the<span class="_ _0"> </span>degree<span class="_ _34"> </span>of<span class="_ _0"> </span><span class="ff3">N<span class="_ _11"> </span></span>is<span class="_ _0"> </span>greater</div><div class="t m0 x80 h3 y479 ff2 fs0 fc0 sc0 ls0 ws0">than<span class="_ _0"> </span>or<span class="_ _34"> </span>equal<span class="_ _0"> </span>to<span class="_ _34"> </span>the<span class="_ _0"> </span>degree<span class="_ _34"> </span>of<span class="_ _0"> </span><span class="ff3">D<span class="_ _3"></span></span>,<span class="_ _34"> </span>divide</div><div class="t m0 x80 h3 y47a ff3 fs0 fc0 sc0 ls0 ws0">N<span class="_ _34"> </span><span class="ff2 lsc">by<span class="_ _0"> </span></span>D<span class="_ _3"></span><span class="ff2">, obtaining</span></div><div class="t m0 x1a5 h3 y47b ff3 fs0 fc0 sc0 ls0 ws0">N<span class="_ _2"></span><span class="ff2">(</span>x<span class="ff2">)</span></div><div class="t m0 x1a5 h3 y41a ff3 fs0 fc0 sc0 ls0 ws0">D<span class="_ _3"></span><span class="ff2">(</span>x<span class="ff2">)</span></div><div class="t m0 x123 h3 yb1 ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff3">Q</span>(<span class="ff3">x</span><span class="ls2">)+</span></div><div class="t m0 x144 h3 y47c ff3 fs0 fc0 sc0 ls0 ws0">N</div><div class="t m0 x1a0 h5 y201 ff8 fs1 fc0 sc0 ls0 ws0"></div><div class="t m0 x14a h3 y47c ff2 fs0 fc0 sc0 ls0 ws0">(<span class="ff3">x</span>)</div><div class="t m0 x95 h3 y41a ff3 fs0 fc0 sc0 ls0 ws0">D<span class="_ _3"></span><span class="ff2">(</span>x<span class="ff2">)</span></div><div class="t m0 xd7 h3 yb1 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x80 h3 yb5 ff2 fs0 fc0 sc0 ls0 ws0">where<span class="_ _7"> </span>the<span class="_ _7"> </span>degree<span class="_ _7"> </span>of<span class="_ _7"> </span><span class="ff3">N</span></div><div class="t m0 x1a2 h5 y9f ff8 fs1 fc0 sc0 ls0 ws0"></div><div class="t m0 xe6 h3 yb5 ff2 fs0 fc0 sc0 ls0 ws0">is<span class="_ _7"> </span>less<span class="_ _7"> </span>than<span class="_ _7"> </span>that<span class="_ _7"> </span>of</div><div class="t m0 x80 h3 yb7 ff3 fs0 fc0 sc0 ls0 ws0">D<span class="_ _3"></span><span class="ff2">.<span class="_ _12"> </span>Second,<span class="_ _0"> </span>factor<span class="_ _0"> </span></span>D<span class="ff2">(</span>x<span class="ff2">).<span class="_ _12"> </span>Use<span class="_ _0"> </span>the<span class="_ _0"> </span>follow-</span></div><div class="t m0 x80 h3 y377 ff2 fs0 fc0 sc0 ls0 ws0">ing rules:<span class="_ _34"> </span>F<span class="_ _5"></span>or a non-repeated factor:</div><div class="t m0 x194 h3 y47d ff3 fs0 fc0 sc0 ls0 ws0">N<span class="_ _2"></span><span class="ff2">(</span>x<span class="ff2">)</span></div><div class="t m0 x8f h4 y47e ff2 fs0 fc0 sc0 ls0 ws0">(<span class="ff3">x<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span>a</span>)<span class="ff3">D</span>(<span class="ff3">x</span>)</div><div class="t m0 x108 h3 y47f ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x8b h3 y480 ff3 fs0 fc0 sc0 ls0 ws0">A</div><div class="t m0 x6 h4 y47e ff3 fs0 fc0 sc0 ls0 ws0">x<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span>a</div><div class="t m0 x124 h3 y47f ff2 fs0 fc0 sc0 ls0 ws0">+</div><div class="t m0 x1ae h3 y480 ff3 fs0 fc0 sc0 ls0 ws0">N</div><div class="t m0 xd7 h5 y376 ff8 fs1 fc0 sc0 ls0 ws0"></div><div class="t m0 x104 h3 y480 ff2 fs0 fc0 sc0 ls0 ws0">(<span class="ff3">x</span>)</div><div class="t m0 xfd h3 y47e ff3 fs0 fc0 sc0 ls0 ws0">D<span class="_ _3"></span><span class="ff2">(</span>x<span class="ff2">)</span></div><div class="t m0 x134 h3 y47f ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x80 h3 y481 ff2 fs0 fc0 sc0 ls0 ws0">where</div><div class="t m0 x100 h3 y482 ff3 fs0 fc0 sc0 ls0 ws0">A<span class="_ _7"> </span><span class="ff2">=</span></div><div class="t m0 x131 h6 y483 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x108 h3 y484 ff3 fs0 fc0 sc0 ls0 ws0">N<span class="_ _2"></span><span class="ff2">(</span>x<span class="ff2">)</span></div><div class="t m0 x108 h3 ycf ff3 fs0 fc0 sc0 ls0 ws0">D<span class="_ _3"></span><span class="ff2">(</span>x<span class="ff2">)</span></div><div class="t m0 x10a h6 y485 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x1af h5 y486 ff6 fs1 fc0 sc0 ls0 ws0">x<span class="ff5">=</span>a</div><div class="t m0 x121 h3 y487 ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x80 h3 y2a7 ff2 fs0 fc0 sc0 ls0 ws0">F<span class="_ _5"></span>o<span class="_ _5"></span>r a repeated factor:</div><div class="t m0 xc7 h3 y488 ff3 fs0 fc0 sc0 ls0 ws0">N<span class="_ _2"></span><span class="ff2">(</span>x<span class="ff2">)</span></div><div class="t m0 x3 h4 y217 ff2 fs0 fc0 sc0 ls0 ws0">(<span class="ff3">x<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span>a</span>)</div><div class="t m0 x86 h5 y489 ff6 fs1 fc0 sc0 ls0 ws0">m</div><div class="t m0 xfc h3 y217 ff3 fs0 fc0 sc0 ls0 ws0">D<span class="_ _3"></span><span class="ff2">(</span>x<span class="ff2">)</span></div><div class="t m0 xe5 h3 y37d ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x13f h5 y194 ff6 fs1 fc0 sc0 ls0 ws0">m<span class="ff8">−<span class="ff5">1</span></span></div><div class="t m0 xf3 h6 yd6 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xf3 h5 y48a ff6 fs1 fc0 sc0 ls0 ws0">k<span class="ff5">=0</span></div><div class="t m0 xd4 h3 y48b ff3 fs0 fc0 sc0 ls0 ws0">A</div><div class="t m0 xd5 h5 y198 ff6 fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 x13e h4 y217 ff2 fs0 fc0 sc0 ls0 ws0">(<span class="ff3">x<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span>a</span>)</div><div class="t m0 x10c h5 y489 ff6 fs1 fc0 sc0 ls0 ws0">m<span class="ff8">−</span>k</div><div class="t m0 x13c h3 y37d ff2 fs0 fc0 sc0 ls0 ws0">+</div><div class="t m0 x134 h3 y48b ff3 fs0 fc0 sc0 ls0 ws0">N</div><div class="t m0 x4 h5 y2a5 ff8 fs1 fc0 sc0 ls0 ws0"></div><div class="t m0 xd9 h3 y48b ff2 fs0 fc0 sc0 ls0 ws0">(<span class="ff3">x</span>)</div><div class="t m0 x1b0 h3 y217 ff3 fs0 fc0 sc0 ls0 ws0">D<span class="_ _3"></span><span class="ff2">(</span>x<span class="ff2">)</span></div><div class="t m0 x16d h3 y37d ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x80 h3 y48c ff2 fs0 fc0 sc0 ls0 ws0">where</div><div class="t m0 x9a h3 y48d ff3 fs0 fc0 sc0 ls0 ws0">A</div><div class="t m0 x1a3 h5 y2e3 ff6 fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 xc8 h3 y48e ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x106 h3 y19b ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 x11f h3 y48f ff3 fs0 fc0 sc0 ls0 ws0">k<span class="_ _3"></span><span class="ff2">!</span></div><div class="t m0 x107 h6 y3d7 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x131 h3 y19b ff3 fs0 fc0 sc0 ls0 ws0">d</div><div class="t m0 x18e h5 y1e3 ff6 fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 xf2 h3 y48f ff3 fs0 fc0 sc0 ls0 ws0">dx</div><div class="t m0 x115 h5 y490 ff6 fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 x142 h6 y3d7 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x13e h3 y19b ff3 fs0 fc0 sc0 ls0 ws0">N<span class="_ _2"></span><span class="ff2">(</span>x<span class="ff2">)</span></div><div class="t m0 x8b h3 y48f ff3 fs0 fc0 sc0 ls0 ws0">D<span class="_ _3"></span><span class="ff2">(</span>x<span class="ff2">)</span></div><div class="t m0 xd5 h6 y3d7 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x133 h5 ye8 ff6 fs1 fc0 sc0 ls0 ws0">x<span class="ff5">=</span>a</div><div class="t m0 x10d h3 y48e ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x80 h3 y103 ff2 fs0 fc0 sc0 ls0 ws0">The<span class="_ _7"> </span>reasonable<span class="_ _7"> </span>man<span class="_ _7"> </span>adapts<span class="_ _7"> </span>himself<span class="_ _7"> </span>to<span class="_ _8"> </span>the</div><div class="t m0 x80 h3 y491 ff2 fs0 fc0 sc0 ls0 ws0">w<span class="_ _5"></span>orld; the<span class="_ _14"> </span>unreasonable persists in<span class="_ _7"> </span>trying</div><div class="t m0 x80 h3 y476 ff2 fs0 fc0 sc0 ls0 ws0">to<span class="_ _0"> </span>adapt<span class="_ _0"> </span>the world<span class="_ _0"> </span>to himself.<span class="_ _11"> </span>Therefore</div><div class="t m0 x80 h3 y2e8 ff2 fs0 fc0 sc0 ls0 ws0">all<span class="_ _7"> </span>progress depends on<span class="_ _7"> </span>the unreasonable.</div><div class="t m0 x80 h3 y492 ff2 fs0 fc0 sc0 ls0 ws0">– George Bernard Sha<span class="_ _5"></span>w</div></div><div class="pi" data-data='{"ctm":[1.673203,0.000000,0.000000,1.673203,0.000000,0.000000]}'></div></div></div>