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 \({a}^{2}-{x}^{2}\), 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
\[\sin^{-1}{(\frac{x}{5})}+C\]

Done