1.40 Remark on substitutions

When doing substitutions from old variable $x$ to new variable $u$, change all the terms involving $x$ into terms involving $u$ in one step; this avoids confusion. If the expressions are complicated, then do the calculations of the derivatives and algebraic simplification separately from the integral before substituting.

Solution.

Let $x=\tan u$, so ${{{dx}\over{du}}=\sec^{2}u;}$ and ${1+x^{2}=1+\tan^{2}u=\sec^{2}u,}$ hence