Integral Formula

Integral formulas on different functions are mentioned here. Apart from the formulas for integration, a brief introduction to integration, integral formulas classification and a few sample questions are also given here.

Integration Definition

Integration can be considered as the reverse process as Differentiation or can be called Inverse Differentiation. Integration is the process of finding a function with its derivative.

Integral Formulas Classification

Integration is the basic operation of Integral Calculus. Integral formulas are classified based on following funtions,

  • Integral formulas based on Rational functions
  • Integral formulas based on Irrational functions
  • Integral formulas based on Trigonometric functions
  • Integral formulas based on Inverse trigonometric functions
  • Integral formulas based on Hyperbolic functions
  • Integral formulas based on Inverse hyperbolic functions
  • Integral formulas based on Exponential functions
  • Integral formulas based on Logarithmic functions
  • Integral formulas based on Gaussian functions

Basic Integration Formulas List

Below is the list of Integral Formulas based on the above functions:

  • $\large \int 1 \; dx = x + C$

  • $\large \int a \; dx = ax + C$

  • $\large \int x^{n} \; dx = \frac{x^{n+1}}{n+1} + C; \; n \neq -1$

  • $\large \int \sin x \; dx = – \cos x + C$

  • $\large \int \cos x \; dx = \sin x + C$

  • $\large \int \sec ^{2} x \; dx = \tan x + C$

  • $\large \int \csc ^{2} x \; dx = -\cot x + C$

  • $\large \int \sec x (\tan x) \; dx = \sec x + C$

  • $\large \int \csc x (\cot x) \; dx = – \csc x + C$

  • $\large \int \frac{1}{x} \; dx = \ln |x| + C$

  • $\large \int e^{x} \; dx = e^{x} + C$

  • $\large \int a^{x} \; dx = \frac{a^{x}}{\ln a} + C; \; a> 0, a\neq 1$

  • $\large \int \frac{1}{\sqrt{1-x^{2}}} \; dx = \sin^{-1} x + C$

  • $\large \int \frac{1}{\sqrt{1+x^{2}}} \; dx = \tan^{-1} x + C$

  • $\large \int \frac{1}{|x|\sqrt{x^{2}-1}} \; dx = \sec^{-1} x + C$

  • $\large \int \sin^{n} (x) dx = \frac{-1}{n} \sin^{n-1} (x) \cos (x) +\frac{n-1}{n} \int \sin^{n-2} (x) dx$

  • $\large \int \cos^{n} (x) dx = \frac{1}{n} \cos^{n-1} (x) \sin (x) +\frac{n-1}{n} \int \cos^{n-2} (x) dx$

  • $\large \int \tan^{n} (x) dx = \frac{1}{n-1} \tan^{n-1} (x) +\int \tan^{n-2} (x) dx$

  • $\large \int \sec^{n} (x) dx = \frac{1}{n-1} \sec^{n-2} (x) \tan (x) + \frac{n-2}{n-1} \int \sec^{n-2} (x) dx$

  • $\large \int \csc^{n} (x) dx = \frac{-1}{n-1} \csc^{n-2} (x) \cot (x) + \frac{n-2}{n-1} \int \csc^{n-2} (x) dx$

Question Based on Integration Formulas

  • Question 1: Calculate \(\int \,5x^{4}\,dx\)
  • Question 2: Find \(\int x\sqrt{1+2x}\;dx\)
  • Question 2: \(\int \frac{1}{x^{2}+6x+25}\,dx\)

Practise This Question

Lump of cotton shrink in water because