Trigonometric Substitution

Reference > Calculus: Integration

 DescriptionA 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 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$Donearcsin(x/5)+C