# Integration Formulas

I have collected some of the most basic and important integration formulas here. These are the ones that you'll get to use every day.

## General Formulas

1. Power of x: $$\int x^n dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1$$
2. Constant: $$\int a \, dx = ax + C$$
3. Exponential: $$\int e^x dx = e^x + C$$
4. a to the power of x: $$\int a^x dx = \frac{a^x}{\ln(a)} + C, \quad a > 0, \, a \neq 1$$
5. Natural Logarithm: $$\int \frac{1}{x} dx = \ln |x| + C$$
6. Product of x and Exponential: $$\int x e^x dx = e^x(x-1) + C$$
7. Summation Power: $$\int { { \left( ax+b \right) }^{ n }dx=\frac { 1 }{ a } } \cdot \frac { { \left( ax+b \right) }^{ n+1 } }{ n+1 } +C$$

## Trigonometric Functions

• $$\int \sin x \, dx = -\cos x + C$$
• $$\int \cos x \, dx = \sin x + C$$
• $$\int \sec^2 x \, dx = \tan x + C$$
• $$\int \csc^2 x \, dx = -\cot x + C$$
• $$\int \sec x \tan x \, dx = \sec x + C$$
• $$\int \csc x \cot x \, dx = -\csc x + C$$

## Inverse Trigonometric Functions

• $$\int \frac{1}{\sqrt{1-x^2}} dx = \sin^{-1} x + C$$
• $$\int \frac{1}{1+x^2} dx = \tan^{-1} x + C$$
• $$\int \frac{1}{x\sqrt{x^2-1}} dx = \sec^{-1} x + C, \quad x > 1$$

## Hyperbolic Functions

• $$\int \sinh x \, dx = \cosh x + C$$
• $$\int \cosh x \, dx = \sinh x + C$$

## Important Integrals

1. Integration by Parts: $$\int u \, dv = uv - \int v \, du$$
2. Trigonometric Identities:
• $$\int \sin^2 x \, dx = \frac{x}{2} - \frac{\sin 2x}{4} + C$$
• $$\int \cos^2 x \, dx = \frac{x}{2} + \frac{\sin 2x}{4} + C$$
3. Exponential Function (General Form): $$\int e^{ax} dx = \frac{1}{a}e^{ax} + C$$
4. Logarithmic Function: $$\int \ln x \, dx = x \ln x - x + C$$
5. Integration of Rational Functions:
• $$\int \frac{1}{(x+a)^n} dx = -\frac{1}{(n-1)(x+a)^{n-1}} + C, \quad n \neq 1$$
6. Arc Length of a Curve: $$\int \sqrt{1 + (\frac{dy}{dx})^2} \, dx$$
7. Area of a Surface of Revolution: $$\int 2\pi y \sqrt{1 + (\frac{dy}{dx})^2} \, dx$$
8. Volume of a Solid of Revolution:
• About the x-axis: $$\int \pi y^2 dx$$
• About the y-axis: $$\int \pi x^2 dy$$