# Power Substitution

## Reference > Calculus: Integration

 DescriptionA 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(4*(5+sqrt(x))^(5/2))/5-(20*(5+sqrt(x))^(3/2))/3+C