What is the integral of ((x-2)sqrt(x+4)) ?

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

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integral of ((x-2)sqrt(x+4))

\int ((x - 2) x + 4) d x

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

Solution steps

\int (x - 2) x + 4 d x

Expand

(x - 2) x + 4 : x x + 4 - 2 x + 4

= \int x x + 4 - 2 x + 4 d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x x + 4 d x - \int 2 x + 4 d x

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

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

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

Add similar elements:

- \frac{8}{3} (x + 4)^{\frac{3}{2}} - \frac{4}{3} (x + 4)^{\frac{3}{2}} = - 4 (x + 4)^{\frac{3}{2}}

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

Add a constant to the solution

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

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