$\int \frac{sin (2 t a n^{- 1} x)}{(1 + x^{2})} d x$

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Solution

$\int \frac{sin (2 {tan}^{- 1} x) d x}{1 + x^{2}}$

put ${tan}^{- 1} x = t$

$\frac{1}{1 + x^{2}} d x = d t$

$\int sin 2 t d t$

$- \frac{1}{2} cos 2 t + C$

$- \frac{1}{2} cos (2 {tan}^{- 1} x) + C$

$- \frac{1}{2} cos (2 {tan}^{- 1} x) + C$ $- \frac{1}{2} cos (2 {tan}^{- 1} x) + C$

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