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Trigonometric SubstitutionDescription A method of integration that uses trigonmetric identities to simplify certain integrals that contain radical expressions. The rules are: If the function contains \({a}^{2}-{x}^{2}\), let \(x=a\sin{u}\) If the function contains \({a}^{2}+{x}^{2}\), let \(x=a\tan{u}\) If the function contains \({x}^{2}-{a}^{2}\), let \(x=a\sec{u}\) |
Examples \[\int \frac{1}{\sqrt{25-{x}^{2}}} \, dx\] 1 Use Trigonometric Substitution Let \(x=5\sin{u}\), \(dx=5\cos{u} \, du\) 2 Substitute variables from above. \[\int \frac{1}{\sqrt{25-{(5\sin{u})}^{2}}}\times 5\cos{u} \, du\] 3 Simplify. \[\int 1 \, du\] 4 Use this rule: \(\int a \, dx=ax+C\). \[u\] 5 From the earlier steps, we know that: \[u=\sin^{-1}{(\frac{1}{5}x)}\] 6 Substitute the above back into the original integral. \[\sin^{-1}{(\frac{1}{5}x)}\] 7 Add constant. \[\sin^{-1}{(\frac{x}{5})}+C\] Done  |
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