| Integrand Contains | Substitution | Based on the Identity |
|---|---|---|
| \(\sqrt{a^{2}-x^{2}}\) | \(x=a\sin \theta\), \(-\dfrac{\pi }{2} \le \theta \, \le \dfrac{\pi }{2}\) | \(1-\sin ^{2}\theta =\cos ^{2}\theta\) |
| \(\sqrt{x^{2}+a^{2}}\) | \(x=a\tan \theta\), \(-\dfrac{\pi }{2} \lt\theta \lt\dfrac{\pi }{2}\) | \(\tan ^{2}\theta +1=\sec ^{2}\theta\) |
| \(\sqrt{x^{2}-a^{2}}\) | \(x=a\sec \theta\), \(0 \le \theta \lt\dfrac{\pi }{2}\), \(\pi \le \theta \lt\dfrac{3\pi }{2}\) | \(\sec ^{2}\theta -1=\tan ^{2}\theta\) |