First slide

Evaluation of definite integrals

Question

The value of the integral
$\int_{- 1}^{1} \frac{d}{d x} (\tan^{- 1} \frac{1}{x}) d x$ is

Moderate

A

$π / 2$
B

$π / 4$
C

$- π / 2$
D

none of these

Solution

$\frac{d}{d x} (\tan^{- 1} \frac{1}{x}) = \frac{1}{1 + \frac{1}{x^{2}}} (- \frac{1}{x^{2}}) = \frac{- 1}{1 + x^{2}}$
$\begin{array}{l} \Rightarrow I = \int_{- 1}^{1} \frac{d}{d x} (\tan^{- 1} \frac{1}{x}) d x = \int_{- 1}^{1} \frac{- d x}{1 + x^{2}} = - {\tan^{- 1} x|}_{- 1}^{1} \\ = - \frac{π}{4} + (- \frac{π}{4}) = - \frac{π}{2} \end{array}$
Note that $I = {\tan^{- 1} (1 / x)|}_{- 1}^{1} = \frac{π}{4} - (- \frac{π}{4}) = \frac{π}{2}$ is incorrect,
since the function $\tan^{- 1} (1 / x)$ is not an antiderivative of $(d / d x) [\tan^{- 1} (1 / x)]$ on the interval
$[- 1, 1]$

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