What is the integral of x(x+1)^{2/3} ?

The integral of x(x+1)^{2/3} is 3/5 x(x+1)^{5/3}-9/40 (x+1)^{8/3}+C

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integral of x(x+1)^{2/3}

\int x (x + 1)^{\frac{2}{3}} d x

\frac{3}{5} x (x + 1)^{\frac{5}{3}} - \frac{9}{40} (x + 1)^{\frac{8}{3}} + C

Solution steps

\int x (x + 1)^{\frac{2}{3}} d x

Apply Integration By Parts

: \frac{3}{5} x (x + 1)^{\frac{5}{3}} - \int \frac{3}{5} (x + 1)^{\frac{5}{3}} d x

= \frac{3}{5} x (x + 1)^{\frac{5}{3}} - \int \frac{3}{5} (x + 1)^{\frac{5}{3}} d x

\int \frac{3}{5} (x + 1)^{\frac{5}{3}} d x = \frac{9}{40} (x + 1)^{\frac{8}{3}}

= \frac{3}{5} x (x + 1)^{\frac{5}{3}} - \frac{9}{40} (x + 1)^{\frac{8}{3}}

Add a constant to the solution

= \frac{3}{5} x (x + 1)^{\frac{5}{3}} - \frac{9}{40} (x + 1)^{\frac{8}{3}} + C

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What is the integral of x(x+1)^{2/3} ?
The integral of x(x+1)^{2/3} is 3/5 x(x+1)^{5/3}-9/40 (x+1)^{8/3}+C