Power Substitution

Reference > Calculus: Integration

Description

A method of integration that aims to replace the nth roots in a function, which are difficult to integrate, with integer powers, which can be easily integrated.

For example, if \(\sqrt[4]{x}\) is in the function, we will let \(x={u}^{4}\).


Examples
\[\int \sqrt{5+\sqrt{x}} \, dx\]
1
Use Power Substitution.
Let \(u=\sqrt{5+\sqrt{x}}\), \(x={u}^{4}-10{u}^{2}+25\), and \(dx=4{u}^{3}-20u \, du\)

2
Expand.
\[\int 4{u}^{4}-20{u}^{2} \, du\]

3
Use Power Rule: \(\int {x}^{n} \, dx=\frac{{x}^{n+1}}{n+1}+C\).
\[\frac{4{u}^{5}}{5}-\frac{20{u}^{3}}{3}\]

4
Substitute \(u=\sqrt{5+\sqrt{x}}\) back into the original integral.
\[\frac{4{\sqrt{5+\sqrt{x}}}^{5}}{5}-\frac{20{\sqrt{5+\sqrt{x}}}^{3}}{3}\]

5
Add constant.
\[\frac{4{(5+\sqrt{x})}^{\frac{5}{2}}}{5}-\frac{20{(5+\sqrt{x})}^{\frac{3}{2}}}{3}+C\]

Done