First slide

Evaluation of definite integrals

Question

The value of $\int_{- 1}^{3} [\tan^{- 1} (\frac{x}{x^{2} + 1}) + \tan^{- 1} (\frac{x^{2} + 1}{x})] dx$ is

Moderate

A

$2 π$
B

$π$
C

$\frac{π}{2}$
D

$\frac{π}{4}$

Solution

$\begin{array}{l} \int_{- 1}^{3} [\tan^{- 1} (\frac{x}{x^{2} + 1}) + \tan^{- 1} (\frac{x^{2} + 1}{x})] dx \\ = \int_{- 1}^{3} [\tan^{- 1} (\frac{x}{x^{2} + 1}) + \cot^{- 1} (\frac{x}{x^{2} + 1})] dx [∵ \tan^{- 1} x = \cot^{- 1} \frac{1}{x}] \\ = \int_{- 1}^{3} (\frac{π}{2}) dx = {[\frac{πx}{2}]}_{- 1}^{3} = \frac{π}{2} [3 + 1] = 2 π \\ [∵ \tan^{- 1} x + \cot^{- 1} x = \frac{π}{2}] \end{array}$

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