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
Let \(u=\sqrt{5+\sqrt{x}}\), \(x={({u}^{2}-5)}^{2}\), and \(dx=4u({u}^{2}-5) du\)

2
Substitute variables from above
\[\int u\times 4u({u}^{2}-5) \,du\]

3
Apply the Constant Factor Rule: \(\int cf(x) \,dx=c\int f(x) \,dx\)
\[4\int {u}^{2}({u}^{2}-5) \,du\]

4
Expand \({u}^{2}({u}^{2}-5)\)
\[4\int {u}^{4}-5{u}^{2} \,du\]

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

6
Expand
\[\frac{4{u}^{5}}{5}-\frac{20{u}^{3}}{3}\]

7
Substitute \(u=\sqrt{5+\sqrt{x}}\) back into the original integral
\[\frac{4{\sqrt{5+\sqrt{x}}}^{5}}{5}-\frac{20{\sqrt{5+\sqrt{x}}}^{3}}{3}\]

8
Add constant
\[\frac{4{(5+\sqrt{x})}^{\frac{5}{2}}}{5}-\frac{20{(5+\sqrt{x})}^{\frac{3}{2}}}{3}+C\]

Done