Integration Formulas
Basic Integration Formulas
1. Constant Rule:
Z
k dx = kx + C
2. Power Rule:
Z
(where k is a constant)
xn+1
+C
n+1
xn dx =
(where n ̸= −1)
3. Exponential Rule:
Z
ex dx = ex + C
4. Trigonometric Functions:
Z
sin(x) dx = − cos(x) + C
Z
cos(x) dx = sin(x) + C
Z
tan(x) dx = ln | sec(x)| + C
5. Inverse Trigonometric Functions:
Z
arcsin(x) dx = x arcsin(x) +
p
arccos(x) dx = x arccos(x) −
p
Z
Z
arctan(x) dx = x arctan(x) −
1 − x2 + C
1 − x2 + C
1
ln |1 + x2 | + C
2
6. Logarithmic Functions:
Z
ln(x) dx = x ln(x) − x + C
7. Rational Functions:
Z
Z
1
dx = ln |x| + C
x
1
x−n+1
+C
dx
=
−
xn
n−1
(where n ̸= 1)
8. Special Forms:
Z
Z
√
1
dx = arcsin(x) + C
1 − x2
1
dx = arctan(x) + C
1 + x2
Additional Integration Formulas
10. Trigonometric Powers (useful in certain physics applications):
Z
Z
1
n−1
sinn (x) dx = − sinn−1 (x) cos(x) +
sinn−2 (x) dx
n
n
Z
Z
n−1
1
n
n−1
cos (x) dx = cos
cosn−2 (x) dx
(x) sin(x) +
n
n
11. Exponential and Trigonometric Functions (useful for oscillatory systems):
Z
ex
ex sin(x) dx = (sin(x) − cos(x)) + C
2
Z
ex
ex cos(x) dx = (sin(x) + cos(x)) + C
2
12. Additional Rational Functions:
Z
Z
1
1
dx = −
+ C (where n ̸= 1)
n
(x − a)
(n − 1)(x − a)n−1
Z
1
1
x−a
dx =
ln
+C
(x − a)(x − b)
b−a
x−b
1
1
x−a
1
x−b
dx =
ln
+
ln
+C
(x − a)(x − b)(x − c)
(b − a)(c − a)
x−b
(a − b)(c − b)
x−c
Z
1
1
dx =
+ C (where m ̸= n)
m
n
(x − a) (x − b)
(m − n)(x − a)m−1 (x − b)n−1
Z
1
x
1
x
1
dx =
ln
−
ln
+C
x(x − a)(x − b)
ab
x−a
ab
x−b
Z
x
a ln |x − b| − b ln |x − a|
x
dx =
−
+C
(x − a)(x − b)
b−a
b−a