Integration by Partial Fraction
If the function to be integrated is of the form f(x) = g(x)/h(x) where g(x) and h(x) are polynomials, then we use the method of partial fraction. There are multiple cases in partial fractions depending upon the type of f(x).
| Type of function f(x) | Partial Fraction |
|---|---|
| [Tex]\frac{px+q}{(x-a)(x-b)} [/Tex]where a ≠ b | [Tex]\frac{A}{x-a}+\frac{B}{x-b} [/Tex] |
| [Tex]\frac{px+q}{(x-a)^2} [/Tex] | [Tex]\frac{A}{x-a}+\frac{B}{(x-a)^2} [/Tex] |
| [Tex]\frac{px^2+qx+r}{(x-a)(x-b)(x-c)} [/Tex] | [Tex]\frac{A}{x-a} + \frac{B}{x-b}+ \frac{C}{x-c} [/Tex] |
| [Tex]\frac{px^2+qx+r}{(x-a)^2(x-b)} [/Tex] | [Tex]\frac{A}{x-a} + \frac{B}{(x-a)^2}+ \frac{C}{x-b} [/Tex] |
| [Tex]\frac{px^2+qx+r}{(x-a)(x^2+bx+c)} [/Tex] | [Tex]\frac{A}{x-a} + \frac{Bx+c}{x^2+bx+c} [/Tex] |
In all these cases, we need to take the LCM of the partial fractions to make the denominator the same. After that, we compare the numerator on the LHS and RHS. Then substitute the suitable value of x in order to make any one part of the numerator zero and determine the value of A, B and C.
Read more about Integration by Partial Fraction.
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
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