What is the integral of (x+5)sqrt(5-x) ?

The integral of (x+5)sqrt(5-x) is -2/3 (x+5)(5-x)^{3/2}-4/15 (5-x)^{5/2}+C

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integral of (x+5)sqrt(5-x)

\int (x + 5) 5 - x d x

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

Solution steps

\int (x + 5) 5 - x d x

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

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What is the integral of (x+5)sqrt(5-x) ?
The integral of (x+5)sqrt(5-x) is -2/3 (x+5)(5-x)^{3/2}-4/15 (5-x)^{5/2}+C