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

The integral of x*sqrt(5-x) is 2/5 (5-x)^{5/2}-10/3 (5-x)^{3/2}+C

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integral of x*sqrt(5-x)

\int x \cdot 5 - x d x

\frac{2}{5} (5 - x)^{\frac{5}{2}} - \frac{10}{3} (5 - x)^{\frac{3}{2}} + C

Solution steps

\int x 5 - x d x

Apply u-substitution

= \int - (- u + 5) u d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int (- u + 5) u d u

Expand

(- u + 5) u : - u^{\frac{3}{2}} + 5 u

= - \int - u^{\frac{3}{2}} + 5 u d u

Apply the Sum Rule:

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

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

\int u^{\frac{3}{2}} d u = \frac{2}{5} u^{\frac{5}{2}}

\int 5 u d u = \frac{10}{3} u^{\frac{3}{2}}

= - (- \frac{2}{5} u^{\frac{5}{2}} + \frac{10}{3} u^{\frac{3}{2}})

Substitute back

u = 5 - x

= - (- \frac{2}{5} (5 - x)^{\frac{5}{2}} + \frac{10}{3} (5 - x)^{\frac{3}{2}})

Simplify

= \frac{2}{5} (5 - x)^{\frac{5}{2}} - \frac{10}{3} (5 - x)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{5} (5 - x)^{\frac{5}{2}} - \frac{10}{3} (5 - x)^{\frac{3}{2}} + C

What is the integral of x*sqrt(5-x) ?
The integral of x*sqrt(5-x) is 2/5 (5-x)^{5/2}-10/3 (5-x)^{3/2}+C