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Question
\[\int\limits_0^1 \tan^{- 1} x dx\]
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Solution
\[\int_0^1 \tan^{- 1} x d x\]
\[ = \int_0^1 \tan^{- 1} x \times 1 d x\]
\[ = \left[ \tan^{- 1} x x \right]_0^1 - \int_0^1 \frac{x}{1 + x^2}dx\]
\[ = \left[ x \tan^{- 1} x \right]_0^1 - \frac{1}{2} \left[ \log\left( 1 + x^2 \right) \right]_0^1 \]
\[ = \frac{\pi}{4} - 0 - \frac{1}{2}\log2 + 0\]
\[ = \frac{\pi}{4} - \frac{1}{2}\log2\]
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