First slide

Evaluation of definite integrals

Question

If $\int_{0}^{\infty} \frac{d x}{(x^{2} + 4) (x^{2} + 9)} = k π$ then the value of $k$ is

Easy

A

1/60
B

1/80
C

1/40
D

1/20

Solution

$\begin{array}{l} \int_{0}^{\infty} \frac{d x}{(x^{2} + 4) (x^{2} + 9)} = \frac{1}{5} \int_{0}^{\infty} [\frac{1}{x^{2} + 4} - \frac{1}{x^{2} + 9}] d x \\ = \frac{1}{5} {[\frac{1}{2} \tan^{- 1} \frac{x}{2} - \frac{1}{3} \tan^{- 1} \frac{x}{3}]}_{0}^{\infty} \\ = \frac{1}{5} [\frac{π}{4} - \frac{π}{6}] = \frac{π}{60}, so k = \frac{1}{60} . \end{array}$

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