Question

\[\int\limits_0^1 \tan^{- 1} \left( \frac{2x}{1 - x^2} \right) dx\]

Answer 1

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