What is the integral of x(x-8)^{1/5} ?

The integral of x(x-8)^{1/5} is 5/6 x(x-8)^{6/5}-25/66 (x-8)^{11/5}+C

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integral of x(x-8)^{1/5}

\int x (x - 8)^{\frac{1}{5}} d x

\frac{5}{6} x (x - 8)^{\frac{6}{5}} - \frac{25}{66} (x - 8)^{\frac{11}{5}} + C

Solution steps

\int x (x - 8)^{\frac{1}{5}} d x

Apply u-substitution

: \int (u + 8) u^{\frac{1}{5}} d u

= \int (u + 8) u^{\frac{1}{5}} d u

Apply Integration By Parts

: \frac{5}{6} u^{\frac{6}{5}} (u + 8) - \int \frac{5}{6} u^{\frac{6}{5}} d u

= \frac{5}{6} u^{\frac{6}{5}} (u + 8) - \int \frac{5}{6} u^{\frac{6}{5}} d u

\int \frac{5}{6} u^{\frac{6}{5}} d u = \frac{25}{66} u^{\frac{11}{5}}

= \frac{5}{6} u^{\frac{6}{5}} (u + 8) - \frac{25}{66} u^{\frac{11}{5}}

Substitute back

u = x - 8

= \frac{5}{6} (x - 8)^{\frac{6}{5}} (x - 8 + 8) - \frac{25}{66} (x - 8)^{\frac{11}{5}}

Simplify

\frac{5}{6} (x - 8)^{\frac{6}{5}} (x - 8 + 8) - \frac{25}{66} (x - 8)^{\frac{11}{5}} : \frac{5}{6} x (x - 8)^{\frac{6}{5}} - \frac{25}{66} (x - 8)^{\frac{11}{5}}

= \frac{5}{6} x (x - 8)^{\frac{6}{5}} - \frac{25}{66} (x - 8)^{\frac{11}{5}}

Add a constant to the solution

= \frac{5}{6} x (x - 8)^{\frac{6}{5}} - \frac{25}{66} (x - 8)^{\frac{11}{5}} + C

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What is the integral of x(x-8)^{1/5} ?
The integral of x(x-8)^{1/5} is 5/6 x(x-8)^{6/5}-25/66 (x-8)^{11/5}+C