First slide

Evaluation of definite integrals

Question

$Evaluate \int_{0}^{\infty} \frac{\tan^{- 1} x}{x (1 + x^{2})} d x$

Moderate

A

$\frac{2}{π} \log 2$
B

$\frac{π}{2} \log 2$
C

$\frac{1}{2} \log 2$
D

$\frac{1}{π} \log 2$

Solution

$\begin{array}{l} I = \int_{0}^{\infty} \frac{\tan^{- 1} x}{x (1 + x^{2})} d x Put x = \tan t so that d x = \sec^{2} t d t . Then, \\ I = \int_{0}^{π / 2} \frac{t}{\tan t \cdot \sec^{2} t} \cdot \sec^{2} t d t = \int_{0}^{π / 2} t \cot t d t = [t \log \sin t]_{0}^{π / 2} - \int_{0}^{π / 2} 1 \cdot \log \sin t d t \\ = - \int_{0}^{π / 2} \log \sin t d t = - (- \frac{π}{2} \log 2) = \frac{π}{2} \log 2 \end{array}$

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