Example of Integration by Partial Fraction
Let us understand partial fractions with an example.
Example: Integrate the function f(x) = x/(x-2)(x+3).
Solution:
Given f(x) = x/(x-2)(x+3)
It is of the form [Tex]f(x) = \frac{px+q}{(x-a)(x-b)} [/Tex]
So it can be written as [Tex]\frac{x}{(x-2)(x+3)} = \frac{A}{x-2}+\frac{B}{x+3} [/Tex]
Taking LCM on RHS, we get,
[Tex]\frac{x}{(x-2)(x+3)} = \frac{A(x+3)+B(x-2)}{(x-2)(x+3)} [/Tex]
Comparing numerators on LHS and RHS:
[Tex]x = A(x+3)+B(x-2) [/Tex]
Put x = 2 to get,
2 = 5A or A = 2/5
similarly put x = -3 to get
-3 = -5B or B = 3/5
Thus [Tex]f(x) = \frac{2}{5(x-2)}+ \frac{3}{5(x+3)} [/Tex]
[Tex]\int f(x) = \int[\frac{2}{5(x-2)}+ \frac{3}{5(x+3)}]dx\\ = \frac{2}{5}\int\frac{1}{x-2} + \frac{3}{5}\int\frac{1}{x+3}\\ = \frac{2}{5}\log{(x-2)} + \frac{3}{5}\log(x+3) [/Tex]
Methods of Integration
Methods of Integration in Calculus refer to the various techniques that are used to integrate function easily. Many times it is not possible to directly integrate a function, so we need to use a specific technique to reduce the integral and then perform integration. Any method of integration involves identifying the type of integral and then deciding which method to use.
In this article, we will study what is Integration in calculus, methods of integration mainly the method of substitution, Integration by parts, and Integration using Trigonometric Identities.
Table of Content
- What is Integration in Calculus?
- What are Methods of Integration?
- Integration by Parts
- Example of Integration by Parts
- Integration By Substitution
- Example of Integration by Substitution
- Integration using Trigonometric Identities
- Example of Integration using Trigonometric Identities
- Integration by Partial Fraction
- Example of Integration by Partial Fraction
- Integration of Some Special Functions
- Important Points related to Methods of Integration
- Examples using Methods of Integration
- Practice Problems on Methods of Integration