What is the integral of x(x-8)^{3/4} ?

The integral of x(x-8)^{3/4} is 4/11 x^2(x-8)^{3/4}-96/77 x(x-8)^{3/4}-1024/77 (x-8)^{3/4}+C

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integral of x(x-8)^{3/4}

\int x (x - 8)^{\frac{3}{4}} d x

\frac{4}{11} x^{2} (x - 8)^{\frac{3}{4}} - \frac{96}{77} x (x - 8)^{\frac{3}{4}} - \frac{1024}{77} (x - 8)^{\frac{3}{4}} + C

Solution steps

\int x (x - 8)^{\frac{3}{4}} d x

Apply Integration By Parts

: \frac{1}{2} x^{2} (x - 8)^{\frac{3}{4}} - \int \frac{3 x ^{2}}{8 ( x - 8 ) ^{\frac{1}{4}}} d x

= \frac{1}{2} x^{2} (x - 8)^{\frac{3}{4}} - \int \frac{3 x ^{2}}{8 ( x - 8 ) ^{\frac{1}{4}}} d x

\int \frac{3 x ^{2}}{8 ( x - 8 ) ^{\frac{1}{4}}} d x = \frac{1}{154} (21 x^{2} + 192 x + 2048) (x - 8)^{\frac{3}{4}}

= \frac{1}{2} x^{2} (x - 8)^{\frac{3}{4}} - \frac{1}{154} (21 x^{2} + 192 x + 2048) (x - 8)^{\frac{3}{4}}

Simplify

\frac{1}{2} x^{2} (x - 8)^{\frac{3}{4}} - \frac{1}{154} (21 x^{2} + 192 x + 2048) (x - 8)^{\frac{3}{4}} : \frac{4}{11} x^{2} (x - 8)^{\frac{3}{4}} - \frac{96}{77} x (x - 8)^{\frac{3}{4}} - \frac{1024}{77} (x - 8)^{\frac{3}{4}}

= \frac{4}{11} x^{2} (x - 8)^{\frac{3}{4}} - \frac{96}{77} x (x - 8)^{\frac{3}{4}} - \frac{1024}{77} (x - 8)^{\frac{3}{4}}

Add a constant to the solution

= \frac{4}{11} x^{2} (x - 8)^{\frac{3}{4}} - \frac{96}{77} x (x - 8)^{\frac{3}{4}} - \frac{1024}{77} (x - 8)^{\frac{3}{4}} + C

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What is the integral of x(x-8)^{3/4} ?
The integral of x(x-8)^{3/4} is 4/11 x^2(x-8)^{3/4}-96/77 x(x-8)^{3/4}-1024/77 (x-8)^{3/4}+C