Question

integration of tan^-1x dx​

Answer 1

$</p><p></p><p>{\color{cyan}Given\; Question \;Is} \\ \\</p><p></p><p>{\color{red}Find\;\;\int \tan^{-1}x \;dx} \\ \\</p><p></p><p>\underline{\underline{\color{cyan}Answer}}\\\\</p><p></p><p></p><p>{\color{green}\int (1*\tan^{-1}x) \;dx} \\ \\</p><p></p><p>\boxed{\color{red} \int u*v \;dx =u\int v\;dv-\int\left(\left(\frac{du}{dv}\right) \int v \;dv\right)du} \\\\</p><p></p><p></p><p>{\color{blue}Take\;\tan^{-1}x \; As \;First \; Function \; And \; 1 \; As \;Second\; Function} \\\\</p><p></p><p></p><p>{\color{green}tan^{-1}x \int 1\;dx -\int\left(\frac{x}{1+x^2}\;dx\right)} \\\\</p><p></p><p></p><p>{\color{green}xtan^{-1}x -\int\frac{1}{2}\left(\frac{2x}{1+x^2}\;dx\right)} \\\\</p><p></p><p></p><p></p><p>{\color{green}xtan^{-1}x -\frac{1}{2}\int\left(\frac{2x}{1+x^2}\;dx\right)} \\\\</p><p></p><p></p><p>{\color{green}xtan^{-1}x -\frac{1}{2}\left(\log(x^2+1)\right) + C} \\\\</p><p></p><p>{\color{cyan}Where\;C\;Is\;Called\;Constant\;OF\; Integration}\\\\</p><p></p><p></p><p>\therefore\;\;{\color{red}\int \tan^{-1}x \;dx} ={\color{green}xtan^{-1}x -\frac{1}{2}\left(\log(x^2+1)\right) + C} \\\\</p><p></p><p>$