What is the integral of (x-3)(sqrt(x+3)) ?

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

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integral of (x-3)(sqrt(x+3))

\int (x - 3) (x + 3) d x

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

Solution steps

\int (x - 3) x + 3 d x

Apply Integration By Parts

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

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

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

Add a constant to the solution

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

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