What is the integral of (x-2)sqrt(8-x) ?

The integral of (x-2)sqrt(8-x) is 2/5 (8-x)^{5/2}-4(8-x)^{3/2}+C

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integral of (x-2)sqrt(8-x)

\int (x - 2) 8 - x d x

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

Solution steps

\int (x - 2) 8 - x d x

Expand

(x - 2) 8 - x : x 8 - x - 2 8 - x

= \int x 8 - x - 2 8 - x 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 8 - x d x - \int 2 8 - x d x

\int x 8 - x d x = \frac{2}{5} (8 - x)^{\frac{5}{2}} - \frac{16}{3} (8 - x)^{\frac{3}{2}}

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

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

Simplify

\frac{2}{5} (8 - x)^{\frac{5}{2}} - \frac{16}{3} (8 - x)^{\frac{3}{2}} - (- \frac{4}{3} (8 - x)^{\frac{3}{2}}) : \frac{2}{5} (8 - x)^{\frac{5}{2}} - 4 (8 - x)^{\frac{3}{2}}

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

Add a constant to the solution

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

What is the integral of (x-2)sqrt(8-x) ?
The integral of (x-2)sqrt(8-x) is 2/5 (8-x)^{5/2}-4(8-x)^{3/2}+C