\[\int x\sqrt[5]{x-1} \, dx\]

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Examples

"factor x^2+5x+6"

"integrate cos(x)^3"

1

Use Power Substitution.

Let

u=\sqrt[5]{x-1}

,

x={u}^{5}+1

, and

dx=5{u}^{4} \, du

2

Substitute variables from above.

\int ({u}^{5}+1)u\times 5{u}^{4} \, du

3

Use Constant Factor Rule:

\int cf(x) \, dx=c\int f(x) \, dx

.

5\int {u}^{5}({u}^{5}+1) \, du

4

Expand.

5\int {u}^{10}+{u}^{5} \, du

5

Use Power Rule:

\int {x}^{n} \, dx=\frac{{x}^{n+1}}{n+1}+C

.

5(\frac{{u}^{11}}{11}+\frac{{u}^{6}}{6})

6

Expand.

5(\frac{{u}^{11}}{11}+\frac{{u}^{6}}{6})

7

Substitute

u=\sqrt[5]{x-1}

back into the original integral.

\frac{5{(\sqrt[5]{x-1})}^{11}}{11}+\frac{5{(\sqrt[5]{x-1})}^{6}}{6}

8

Add constant.

\frac{5{(x-1)}^{\frac{11}{5}}}{11}+\frac{5{(x-1)}^{\frac{6}{5}}}{6}+C

Done

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