A table of indefinite integrals of functions is presented below.
In what follows, \( c \) is a constant of integration and can take any constant value.
1 - Integrals of Elementary Functions
- 1.1 \( \displaystyle \int dx = x + c \)
- 1.2 \( \displaystyle \int k \, dx = kx + c \), where \( k \) is a constant.
- 1.3 \( \displaystyle \int x^n \, dx = \dfrac{x^{n+1}}{n+1} + c \) (\( n \neq -1 \))
- 1.4 \( \displaystyle \int \dfrac{1}{x} \, dx = \ln|x| + c \)
2 - Integrals of Elementary Trigonometric Functions
Involving \( \sin x \), \( \cos x \), \( \tan x \), \( \cot x \), \( \sec x \), and \( \csc x \).
- 2.1 \( \displaystyle \int \sin x \, dx = -\cos x + c \)
- 2.2 \( \displaystyle \int \cos x \, dx = \sin x + c \)
- 2.3 \( \displaystyle \int \tan x \, dx = \ln|\sec x| + c \)
- 2.4 \( \displaystyle \int \cot x \, dx = \ln|\sin x| + c \)
- 2.5 \( \displaystyle \int \sec x \, dx = \ln|\sec x + \tan x| + c \)
- 2.6 \( \displaystyle \int \csc x \, dx = \ln|\csc x - \cot x| + c \)
3 - Integrals Involving More Than One Trigonometric Function
- 3.1 \( \displaystyle \int \sec x \tan x \, dx = \sec x + c \)
- 3.2 \( \displaystyle \int \csc x \cot x \, dx = -\csc x + c \)
- 3.3 \( \displaystyle \int \sin mx \sin nx \, dx = -\dfrac{\sin[(m+n)x]}{2(m+n)} + \dfrac{\sin[(m-n)x]}{2(m-n)} + c \), for \( m \neq n \)
- 3.4 \( \displaystyle \int \cos mx \cos nx \, dx = \dfrac{\sin[(m+n)x]}{2(m+n)} + \dfrac{\sin[(m-n)x]}{2(m-n)} + c \), for \( m \neq n \)
- 3.5 \( \displaystyle \int \sin mx \cos nx \, dx = -\dfrac{\cos[(m+n)x]}{2(m+n)} - \dfrac{\cos[(m-n)x]}{2(m-n)} + c \), for \( m \neq n \)
4 - Integrals Involving Exponential and Logarithmic Functions
- 4.1 \( \displaystyle \int e^x \, dx = e^x + c \)
- 4.2 \( \displaystyle \int a^x \, dx = \dfrac{a^x}{\ln a} + c \)
- 4.3 \( \displaystyle \int \ln x \, dx = x \ln x - x + c \)
5 - Integrals of Inverse Trigonometric Functions
Involving \( \arcsin x \), \( \arccos x \), \( \arctan x \), \( \text{arccot} \, x \), \( \text{arcsec} \, x \), and \( \text{arccsc} \, x \).
- 5.1 \( \displaystyle \int \arcsin x \, dx = x \arcsin x + \sqrt{1 - x^2} + c \)
- 5.2 \( \displaystyle \int \arccos x \, dx = x \arccos x - \sqrt{1 - x^2} + c \)
- 5.3 \( \displaystyle \int \arctan x \, dx = x \arctan x - \ln\left|\sqrt{1 + x^2}\right| + c \)
- 5.4 \( \displaystyle \int \text{arccot} \, x \, dx = x \, \text{arccot} \, x + \ln\sqrt{1 + x^2} + c \)
- 5.5 \( \displaystyle \int \text{arcsec} \, x \, dx = x \, \text{arcsec} \, x - \ln\left|x + \sqrt{x^2 - 1}\right| + c \)
- 5.6 \( \displaystyle \int \text{arccsc} \, x \, dx = x \, \text{arccsc} \, x + \ln\left|x + \sqrt{x^2 - 1}\right| + c \)
6 - Integrals Involving Exponential and Sine/Cosine Functions
- 6.1 \( \displaystyle \int e^{ax} \sin bx \, dx = \dfrac{e^{ax}}{a^2 + b^2} (a \sin bx - b \cos bx) + c \)
- 6.2 \( \displaystyle \int e^{ax} \cos bx \, dx = \dfrac{e^{ax}}{a^2 + b^2} (b \sin bx + a \cos bx) + c \)
7 - Integrals Involving Hyperbolic Functions
Involving \( \sinh x \), \( \cosh x \), \( \tanh x \), \( \coth x \), \( \text{sech} \, x \), and \( \text{csch} \, x \).
- 7.1 \( \displaystyle \int \sinh x \, dx = \cosh x + c \)
- 7.2 \( \displaystyle \int \cosh x \, dx = \sinh x + c \)
- 7.3 \( \displaystyle \int \text{sech} \, x \tanh x \, dx = -\text{sech} \, x + c \)
- 7.4 \( \displaystyle \int \text{csch} \, x \coth x \, dx = -\text{csch} \, x + c \)
- 7.5 \( \displaystyle \int \text{sech}^2 x \, dx = \tanh x + c \)
- 7.6 \( \displaystyle \int \text{csch}^2 x \, dx = -\coth x + c \)