Question

Determine the integral of tan inverse x dx

Answer 1

You mean \[\int\arctan{x}dx?\]In that case,\begin{eqnarray*}
\int\arctan{x}dx&=&\int (x')\arctan{x}dx \\&=&x \arctan{x} - \int\frac{x}{1+x^2} dx \\&=&x \arctan{x}-\frac{\log(1+x^2)}{2} + C.\end{eqnarray*}

Answer 2

love that integration by parts!