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

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

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

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

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

Solution steps

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

Expand

(a - x) a + x : a a + x - x a + x

= \int a a + x - x a + 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 a a + x d x - \int x a + x d x

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

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

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

Simplify

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

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

Add a constant to the solution

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

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