The value of the integral $\int\limits_{1/2}^2 {{{{{\tan }^{ - 1}}x} \over x}dx}$ is equal to :
Solution
<p>$I = \int\limits_{{1 \over 2}}^2 {{{{{\tan }^{ - 1}}x} \over x}dx}$ ..... (i)</p>
<p>$x \to {1 \over x}$</p>
<p>$I = \int\limits_{{1 \over 2}}^2 {{1 \over x}{{\tan }^{ - 1}}{1 \over x}dx}$ ..... (ii)</p>
<p>$2I = \int\limits_{{1 \over 2}}^2 {{1 \over x}\,.\,{\pi \over 2}dx}$</p>
<p>$= \left. {{\pi \over 2}\ln x} \right|_{{1 \over 2}}^2 = \pi \ln 2$</p>
<p>$\Rightarrow I = {\pi \over 2}\ln 2$</p>
About this question
Subject: Mathematics · Chapter: Definite Integration · Topic: Properties of Definite Integrals
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