Theoretical Computer Science Cheat Sheet
Calculus Cont.
15.
arccos
x
a
dx = arccos
x
a
−
a
2
− x
2
,a>0, 16.
arctan
x
a
dx = x arctan
x
a
−
a
2
ln(a
2
+ x
2
),a>0,
17.
sin
2
(ax)dx =
1
2a
ax − sin(ax) cos(ax)
, 18.
cos
2
(ax)dx =
1
2a
ax + sin(ax) cos(ax)
,
19.
sec
2
xdx= tan x, 20.
csc
2
xdx= −cot x,
21.
sin
n
xdx= −
sin
n−1
x cos x
n
+
n − 1
n
sin
n−2
x dx, 22.
cos
n
xdx=
cos
n−1
x sin x
n
+
n − 1
n
cos
n−2
x dx,
23.
tan
n
xdx=
tan
n−1
x
n − 1
−
tan
n−2
x dx, n =1, 24.
cot
n
xdx= −
cot
n−1
x
n − 1
−
cot
n−2
x dx, n =1,
25.
sec
n
xdx=
tan x sec
n−1
x
n − 1
+
n − 2
n − 1
sec
n−2
x dx, n =1,
26.
csc
n
xdx= −
cot x csc
n−1
x
n − 1
+
n − 2
n − 1
csc
n−2
x dx, n =1, 27.
sinh xdx= cosh x, 28.
cosh xdx= sinh x,
29.
tanh xdx=ln|cosh x|, 30.
coth xdx=ln|sinh x|, 31.
sech xdx= arctan sinhx, 32.
csch xdx=ln
tanh
x
2
,
33.
sinh
2
xdx=
1
4
sinh(2x) −
1
2
x, 34.
cosh
2
xdx=
1
4
sinh(2x)+
1
2
x, 35.
sech
2
xdx= tanh x,
36.
arcsinh
x
a
dx = x arcsinh
x
a
−
x
2
+ a
2
,a>0, 37.
arctanh
x
a
dx = x arctanh
x
a
+
a
2
ln |a
2
− x
2
|,
38.
arccosh
x
a
dx =
x arccosh
x
a
−
x
2
+ a
2
, if arccosh
x
a
> 0 and a>0,
x arccosh
x
a
+
x
2
+ a
2
, if arccosh
x
a
< 0 and a>0,
39.
dx
√
a
2
+ x
2
=ln
x +
a
2
+ x
2
,a>0,
40.
dx
a
2
+ x
2
=
1
a
arctan
x
a
,a>0, 41.
a
2
− x
2
dx =
x
2
a
2
− x
2
+
a
2
2
arcsin
x
a
,a>0,
42.
(a
2
− x
2
)
3/2
dx =
x
8
(5a
2
− 2x
2
)
a
2
− x
2
+
3a
4
8
arcsin
x
a
,a>0,
43.
dx
√
a
2
− x
2
= arcsin
x
a
,a>0, 44.
dx
a
2
− x
2
=
1
2a
ln
a + x
a − x
, 45.
dx
(a
2
− x
2
)
3/2
=
x
a
2
√
a
2
− x
2
,
46.
a
2
± x
2
dx =
x
2
a
2
± x
2
±
a
2
2
ln
x +
a
2
± x
2
, 47.
dx
√
x
2
− a
2
=ln
x +
x
2
− a
2
,a>0,
48.
dx
ax
2
+ bx
=
1
a
ln
x
a + bx
, 49.
x
√
a + bx dx =
2(3bx − 2a)(a + bx)
3/2
15b
2
,
50.
√
a + bx
x
dx =2
√
a + bx + a
1
x
√
a + bx
dx, 51.
x
√
a + bx
dx =
1
√
2
ln
√
a + bx −
√
a
√
a + bx +
√
a
,a>0,
52.
√
a
2
− x
2
x
dx =
a
2
− x
2
− a ln
a +
√
a
2
− x
2
x
, 53.
x
a
2
− x
2
dx = −
1
3
(a
2
− x
2
)
3/2
,
54.
x
2
a
2
− x
2
dx =
x
8
(2x
2
− a
2
)
a
2
− x
2
+
a
4
8
arcsin
x
a
,a>0, 55.
dx
√
a
2
− x
2
= −
1
a
ln
a +
√
a
2
− x
2
x
,
56.
xdx
√
a
2
− x
2
= −
a
2
− x
2
, 57.
x
2
dx
√
a
2
− x
2
= −
x
2
a
2
− x
2
+
a
2
2
arcsin
x
a,
a>0,
58.
√
a
2
+ x
2
x
dx =
a
2
+ x
2
− a ln
a +
√
a
2
+ x
2
x
, 59.
√
x
2
− a
2
x
dx =
x
2
− a
2
− a arccos
a
|x|
,a>0,
60.
x
x
2
± a
2
dx =
1
3
(x
2
± a
2
)
3/2
, 61.
dx
x
√
x
2
+ a
2
=
1
a
ln
x
a +
√
a
2
+ x
2
,