Integral Formulas

Integral Formulas – Basic Integration Formula on different functions are mentioned here. Apart from the formulas for integration, a brief introduction to integration, classification of formulas and a few sample questions are also given here, which you can practice based on the integration formulas mentioned in this article. These integral formulas are equally important as differentiation formulas. When we speak about integration by parts, it is with regard to integrating the product of two functions, say y = uv. Some more concepts related to integral you will learn with us here, so keep learning. Also, watch the video given below to clear your concept.

Integration Definition

Integration can be considered as the reverse process of Differentiation or can be called Inverse Differentiation. Integration is the process of finding a function with its derivative. It is used to find many mathematical quantities such as areas, volumes, displacement, etc. on a small scale to large scale industries. There are basically two types of Integrals; Definite and Indefinite.

Classification

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

  • Rational functions
  • Irrational functions
  • Trigonometric functions
  • Inverse trigonometric functions
  • Hyperbolic functions
  • Inverse hyperbolic functions
  • Exponential functions
  • Logarithmic functions
  • Gaussian functions

List of Integral Formulas

Below is the list of integral formulas, based on the above functions:

  • $\large \int 1 \; dx = x + C$

  • $\large \int a \; dx = ax + C$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Solve Using Integral Formulas

1. Calculate \(\int \,5x^{4}\,dx\)

2. Find \(\int x\sqrt{1+2x}\;dx\)

3. Solve\(\int \frac{1}{x^{2}+6x+25}\,dx\)

Practise This Question

What is an aperture in the lens?

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