Trigonometric Substitution

Reference > Calculus: Integration

Description

A method of integration that uses trigonmetric identities to simplify certain integrals that contain radical expressions. The rules are:

If the function contains a2−x2{a}^{2}-{x}^{2}, let x=asin⁡ux=a\sin{u}

If the function contains a2+x2{a}^{2}+{x}^{2}, let x=atan⁡ux=a\tan{u}

If the function contains x2−a2{x}^{2}-{a}^{2}, let x=asec⁡ux=a\sec{u}


Examples
∫125−x2 dx\int \frac{1}{\sqrt{25-{x}^{2}}} \, dx
1
Use Trigonometric Substitution
Let x=5sin⁡ux=5\sin{u}, dx=5cos⁡u dudx=5\cos{u} \, du

2
Substitute variables from above.
∫125−(5sin⁡u)2×5cos⁡u du\int \frac{1}{\sqrt{25-{(5\sin{u})}^{2}}}\times 5\cos{u} \, du

3
Simplify.
∫1 du\int 1 \, du

4
Use this rule: ∫a dx=ax+C\int a \, dx=ax+C.
uu

5
From the earlier steps, we know that:
u=sin⁡−1(15x)u=\sin^{-1}{(\frac{1}{5}x)}

6
Substitute the above back into the original integral.
sin⁡−1(15x)\sin^{-1}{(\frac{1}{5}x)}

7
Add constant.
sin⁡−1(x5)+C\sin^{-1}{(\frac{x}{5})}+C

Done

See Also

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