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 x4\sqrt[4]{x} is in the function, we will let x=u4x={u}^{4}.


Examples
∫5+x dx\int \sqrt{5+\sqrt{x}} \, dx
1
Use Power Substitution.
Let u=5+xu=\sqrt{5+\sqrt{x}}, x=u4−10u2+25x={u}^{4}-10{u}^{2}+25, and dx=4u3−20u dudx=4{u}^{3}-20u \, du

2
Expand.
∫4u4−20u2 du\int 4{u}^{4}-20{u}^{2} \, du

3
Use Power Rule: ∫xn dx=xn+1n+1+C\int {x}^{n} \, dx=\frac{{x}^{n+1}}{n+1}+C.
4u55−20u33\frac{4{u}^{5}}{5}-\frac{20{u}^{3}}{3}

4
Substitute u=5+xu=\sqrt{5+\sqrt{x}} back into the original integral.
45+x55−205+x33\frac{4{\sqrt{5+\sqrt{x}}}^{5}}{5}-\frac{20{\sqrt{5+\sqrt{x}}}^{3}}{3}

5
Add constant.
4(5+x)525−20(5+x)323+C\frac{4{(5+\sqrt{x})}^{\frac{5}{2}}}{5}-\frac{20{(5+\sqrt{x})}^{\frac{3}{2}}}{3}+C

Done

See Also

Cymath foriOSAndroid