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1.40 Remark on substitutions

When doing substitutions from old variable xx to new variable uu, change all the terms involving xx into terms involving uu 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.

Example.

Evaluate I=01(1+x2)-3/2dx.I=\int_{0}^{1}(1+x^{2})^{-3/2}dx.

Solution.

Let x=tanux=\tan u, so dxdu=sec2u;{{{dx}\over{du}}=\sec^{2}u;} and 1+x2=1+tan2u=sec2u,{1+x^{2}=1+\tan^{2}u=\sec^{2}u,} hence

I=0π4(sec2u)-3/2sec2udu=0π4cosudu 1/2.{I=\int_{0}^{\frac{\pi}{4}}(\sec^{2}u)^{-3/2}\sec^{2}u\;du}\,{=\int_{0}^{\frac% {\pi}{4}}\cos u\;du}\,{1/\sqrt{2}.}