Integration by Partial Fractions


  • Integrals which are in the form
    \int \frac{f(x)}{g(x)} dx (Where degree of f(x) < degree of g(x))
  • Process : using the process of partial fractions by the following procedure
    \frac{1}{(x+a)(x+b)} = \frac{A}{x+a} + \frac{B}{x+b}
    \frac{1}{(x+a)^{2} (x+b)} = \frac{A}{(x+a)} + \frac{B}{(x+a)^{2}} + \frac{C}{(x+b)}
    \frac{1}{(x+a)(x^{2}+b)} = \frac{A}{x+a} + \frac{Bx+c}{x^{2}+b} 

View the Topic in this video From 13:35 To 48:50

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S.No Form of the rational function Form of the partial fraction
1. \frac{px+q}{(x-a)(x-b)},a\neq b  \frac{A}{x-a}+\frac{B}{x-b}
2. \frac{px+q}{(x-a)^2}  \frac{A}{x-a}+\frac{B}{(x-a)^2}
3. \frac{px^2+qx+r}{(x-a)(x-b)(x-c)}  \frac{A}{x-a}+\frac{B}{x-b}+\frac{C}{x-c}
4. \frac{px^2+qx+r}{(x-a)^2(x-b)}  \frac{A}{x-a}+\frac{B}{(x-a)^2}+\frac{C}{x-b}
5. \frac{px^2+qx+r}{(x-a)(x^2+bx+c)}  \frac{A}{x-a}+\frac{Bx+C}{x^2+bx+c}

 where x2 + bx +c cannot be factorised further