Basic Integration Formulas

Basic Integration Formulas

• Constant Function:
  $\int k \, dx = kx + C$ where $k$ is any constant.

• Power Rule:
  $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ where $n \neq -1$.

• Reciprocal Function:
  $\int \frac{1}{x} \, dx = \ln|x| + C$

• Exponential Function (base e):
  $\int e^x \, dx = e^x + C$

• Exponential Function (base a):
  $\int a^x \, dx = \frac{a^x}{\ln a} + C$ where $a > 0$ and $a \neq 1$.

• Sine Function:
  $\int \sin x \, dx = -\cos x + C$

• Cosine Function:
  $\int \cos x \, dx = \sin x + C$

• Secant Squared Function:
  $\int \sec^2 x \, dx = \tan x + C$

• Cosecant Squared Function:
  $\int \csc^2 x \, dx = -\cot x + C$

• Secant x Tangent x Function:
  $\int \sec x \tan x \, dx = \sec x + C$

• Cosecant x Cotangent x Function:
  $\int \csc x \cot x \, dx = -\csc x + C$

• Tangent Function:
  $\int \tan x \, dx = \ln |\sec x| + C$

• Cotangent Function:
  $\int \cot x \, dx = \ln |\sin x| + C$

• Secant Function:
  $\int \sec x \, dx = \ln |\sec x + \tan x| + C$

• Cosecant Function:
  $\int \csc x \, dx = -\ln |\csc x + \cot x| + C$

• Hyperbolic Sine Function:
  $\int \sinh x \, dx = \cosh x + C$

• Hyperbolic Cosine Function:
  $\int \cosh x \, dx = \sinh x + C$

• Inverse Sine Function:
  $\int \frac{dx}{\sqrt{a^2 - x^2}} = \arcsin(\frac{x}{a}) + C$

• Inverse Tangent Function:
  $\int \frac{dx}{a^2 + x^2} = \frac{1}{a} \arctan(\frac{x}{a}) + C$

• Inverse Secant Function:
  \int \frac{dx}{x\sqrt{x^2 - a^2}} = \frac{1}{a} \arcsec|\frac{x}{a}| + C

• Inverse Hyperbolic Sine Function:
  $\int \frac{dx}{\sqrt{a^2 + x^2}} = \sinh^{-1}(\frac{x}{a}) + C$

• Inverse Hyperbolic Cosine Function:
  $\int \frac{dx}{\sqrt{x^2 - a^2}} = \cosh^{-1}(\frac{x}{a}) + C$ where $x > a > 0$