What is the integral of x^2(5x+8)^{1/2} ?

The integral of x^2(5x+8)^{1/2} is 2/15 x^2(5x+8)^{3/2}-4/375 (2/7 (5x+8)^{7/2}-16/5 (5x+8)^{5/2})+C

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integral of x^2(5x+8)^{1/2}

\int x^{2} (5 x + 8)^{\frac{1}{2}} d x

\frac{2}{15} x^{2} (5 x + 8)^{\frac{3}{2}} - \frac{4}{375} (\frac{2}{7} (5 x + 8)^{\frac{7}{2}} - \frac{16}{5} (5 x + 8)^{\frac{5}{2}}) + C

Solution steps

\int x^{2} (5 x + 8)^{\frac{1}{2}} d x

Apply Integration By Parts

: \frac{2}{15} x^{2} (5 x + 8)^{\frac{3}{2}} - \int \frac{4}{15} x (5 x + 8)^{\frac{3}{2}} d x

= \frac{2}{15} x^{2} (5 x + 8)^{\frac{3}{2}} - \int \frac{4}{15} x (5 x + 8)^{\frac{3}{2}} d x

\int \frac{4}{15} x (5 x + 8)^{\frac{3}{2}} d x = \frac{4}{375} (\frac{2}{7} (5 x + 8)^{\frac{7}{2}} - \frac{16}{5} (5 x + 8)^{\frac{5}{2}})

= \frac{2}{15} x^{2} (5 x + 8)^{\frac{3}{2}} - \frac{4}{375} (\frac{2}{7} (5 x + 8)^{\frac{7}{2}} - \frac{16}{5} (5 x + 8)^{\frac{5}{2}})

Add a constant to the solution

= \frac{2}{15} x^{2} (5 x + 8)^{\frac{3}{2}} - \frac{4}{375} (\frac{2}{7} (5 x + 8)^{\frac{7}{2}} - \frac{16}{5} (5 x + 8)^{\frac{5}{2}}) + C

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What is the integral of x^2(5x+8)^{1/2} ?
The integral of x^2(5x+8)^{1/2} is 2/15 x^2(5x+8)^{3/2}-4/375 (2/7 (5x+8)^{7/2}-16/5 (5x+8)^{5/2})+C