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.

You can read it online or download the integration formulas PDF here.

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: $$\int \pi x^2 dy$$