First slide

Methods of integration

Question

$E v a l u a t e \int \frac{x^{2}}{(x^{2} + 1) (x^{2} + 4)} d x$

Moderate

A

$- \frac{1}{3} \tan^{- 1} x + \frac{4}{3} \tan^{- 1} (\frac{x}{2}) + C$
B

$- \frac{1}{3} \tan^{- 1} x + \frac{2}{3} \tan^{- 1} (\frac{x}{2}) + C$
C

$\frac{1}{3} \tan^{- 1} x + \frac{2}{3} \tan^{- 1} (\frac{x}{2}) + C$
D

None of these

Solution

$\int \frac{x^{2}}{(x^{2} + 1) (x^{2} + 4)} d x = \frac{1}{3} \int [\frac{4}{x^{2} + 4} - \frac{1}{x^{2} + 1}] d x = - \frac{1}{3} \int \frac{1}{x^{2} + 1} d x + \frac{4}{3} \int \frac{1}{x^{2} + 4} d x = - \frac{1}{3} \tan^{- 1} x + \frac{4}{3} \cdot \frac{1}{2} \tan^{- 1} (\frac{x}{2}) + C = - \frac{1}{3} \tan^{- 1} x + \frac{2}{3} \tan^{- 1} (\frac{x}{2}) + C$

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