Step By Step Calculus » 15.5 - Trigonometric versus Polynomial Substitutions

Synopsis
Trigonometric substitution plays an important role in evaluating integrals involving any of these terms: \sqrt{a^2-b^2x^2}\sqrt{a^2-b^2x^2}, \sqrt{a^2+b^2x^2}\sqrt{a^2+b^2x^2},and \sqrt{b^2x^2-a^2}\sqrt{b^2x^2-a^2}. Generally, one can use the following guidelines for substitution in the integrals involving one of those terms.
Trigonometric Substitution: (Category 1)
\begin{tabular}{l|l|l} \hline Form of integral involving $\;$& \multicolumn{1}{c}{Substitution} & \multicolumn{1}{|c}{Identities to use} \\ \multicolumn{1}{c|}{$\displaystyle \sqrt{a^2 -b^2x^2}$} & & \\ \hline $\displaystyle \int \left( \sqrt{a^2-b^2x^2}\right)^mx^{2n+1}dx$ & Use $% u=a^2-b^2x^2$ & $\displaystyle \left( a^2-u\right)^n=\sum_{k=0}^n {\binom{n}{% k}}\left( a^2\right)^{n-k}(-u)^k$ \\ for $a,b>0,\;n\in \mathbb{N}$ and $m\in \mathbb{Z}$ & & \\ \hline other & Try $\displaystyle x=\frac{a}{b}\sin \theta,$ & $\displaystyle \sqrt{a^2-a^2\sin^2\theta} = a|\cos \theta|$ \\ & with $\displaystyle \theta \in \left[ \frac{-\pi}{2},\frac{\pi}{2}\right]$ & $=a\cos \theta$ since $\displaystyle \theta \in \left[ \frac{-\pi}{2},% \frac{\pi}{2}\right]$ \\ \hline \end{tabular}
Trigonometric Substitution: (Category 2)
\begin{tabular}{l|l|l} \hline Form of integral involving & \multicolumn{1}{c}{Substitution} & \multicolumn{1}{|c}{Identities to use} \\ \multicolumn{1}{c|}{$\displaystyle \sqrt{a^2 +b^2x^2}$} & & \\ \hline $\displaystyle \int \left( \sqrt{a^2+b^2x^2}\right)^mx^{2n+1}dx$ & Use $% u=a^2+b^2x^2$ & $\displaystyle \left( u-a^2\right)^n=\sum_{k=0}^n {\binom{n}{% k}}u^{n-k}\left( -a^2\right)^k$ \\ for $a,b>0,\;n\in \mathbb{N}$ and $m\in \mathbb{Z}$ & & \\ \hline other & Try $\displaystyle x=\frac{a}{b}\tan \theta,$ & $\displaystyle \sqrt{a^2+a^2\tan^2\theta} = a|\sec \theta|$ \\ & with $\displaystyle \theta \in \left( \frac{-\pi}{2},\frac{\pi}{2}\right)$ & $=a\sec \theta$ since $\displaystyle \theta \in \left( \frac{-\pi}{2},% \frac{\pi}{2}\right)$ \\ \hline \end{tabular}
Trigonometric Substitution: (Category 3)
\begin{tabular}{l|l|l} \hline Form of integral involving & \multicolumn{1}{c}{Substitution} & \multicolumn{1}{|c}{Identities to use} \\ \multicolumn{1}{c|}{$\displaystyle \sqrt{b^2x^2-a^2}$} & & \\ \hline $\displaystyle \int \left( \sqrt{b^2x^2-a^2}\right)^mx^{2n+1}dx$ & Use $% u=b^2x^2-a^2$ & $\displaystyle \left( a^2+u\right)^n=\sum_{k=0}^n {\binom{n}{% k}}\left( a^2\right)^{n-k}u^k$ \\ for $a,b>0,\;n\in \mathbb{N}$ and $m\in \mathbb{Z}$ & & \\ \hline other & Try $\displaystyle x=\frac{a}{b}\sec \theta,$ & $\displaystyle \sqrt{a^2\sec^2\theta-a^2} = a|\tan \theta|$ \\ & with $\displaystyle \theta \in \left[ 0,\frac{\pi}{2}\right)\cup\left(\frac{\pi}{2},\pi\right]$ & $\displaystyle=\left\{\begin{array}{ll}a\tan\theta & \textrm{ when } \theta\in\left[ 0,\frac{\pi}{2}\right)\\ -a\tan\theta & \textrm{ when } \theta \in\left(\frac{\pi}{2},\pi\right]\end{array}\right.$\\ \hline \end{tabular}