Table of Integral Formulas

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 $$\displaystyle \int k \; dx = k x + c$$, where $$k$$ is a constant.
1.3 $$\displaystyle \int x^n \; dx = \dfrac{x^{n+1}}{n+1} + c$$
1.4 $$\displaystyle \int \dfrac{1}{x} \; dx = \ln |x| + c$$

2 - Integrals of Elementary Trigonometric Functions : $$\sin x$$, $$\cos x$$, $$\tan x$$, $$\cot x$$, $$\sec x$$ 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 $$\displaystyle \int \sin mx \sin nx \; dx = -\dfrac{\sin[(m+n)x]}{2(m+n)} + \dfrac{\sin[(m-n)x]}{2(m-n)} + c$$,
with $$m \neq n$$
3.4 $$\displaystyle \int \cos mx \cos nx \; dx = \dfrac{\sin[(m+n)x]}{2(m+n)} + \dfrac{\sin[(m-n)x]}{2(m-n)} + c$$,
with $$m \neq n$$
3.5 $$\displaystyle \int \sin mx \cos nx \; dx = -\dfrac{\cos[(m+n)x]}{2(m+n)} - \dfrac{\cos[(m-n)x]}{2(m-n)} + c$$,
with $$m \neq n$$

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: $$\arcsin x$$, $$\arccos x$$, $$\arctan x$$, $$\text{arccot} \; x$$, $$\text{arcsec}\; x$$ 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 and 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: $$\sinh x$$, $$\cosh x$$, $$\tanh x$$, $$\coth x$$, $$\text{sech} \; x$$, $$\text{csch} \; x$$.

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 $$\displaystyle \int \text{csch}^2 x \; dx = - \;\coth x + c$$
Más referencias sobre
integrales y sus aplicaciones en cálculo.