QJEEM23Madc75: JEE Main 2023 Mathematics - Class 12 JEEM23Madc75 Solution: The value of the integral ∫ 1 / 2 2 tan − 1 x x d x is equal to :

Question

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The value of the integral $\int_{1 / 2}^{2} \frac{\tan^{- 1} x}{x} d x$ is equal to :

Accepted Answer

$I = \int_{\frac{1}{2}}^{2} \frac{\tan^{- 1} x}{x} d x$ ..... (i)

$x \to \frac{1}{x}$

$I = \int_{\frac{1}{2}}^{2} \frac{1}{x} \tan^{- 1} \frac{1}{x} d x$ ..... (ii)

$2 I = \int_{\frac{1}{2}}^{2} \frac{1}{x} . \frac{π}{2} d x$

$= {\frac{π}{2} \ln x |}_{\frac{1}{2}}^{2} = π \ln 2$

$\Rightarrow I = \frac{π}{2} \ln 2$

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