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

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

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integral of (4-x)sqrt(x)

\int (4 - x) x d x

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

Solution steps

\int (4 - x) x d x

Expand

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

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

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

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

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

Add a constant to the solution

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

\int \frac{3 x}{x ^{3} - 1} d x

\int \frac{x - 4}{x + 2} d x

\int \frac{2 x ^{2}}{x ^{2} + 2} d x

\int \frac{t}{3 t ^{2} + 3} d t

\frac{3}{x}

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