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

The integral of sqrt(x)(x-2)^2 is 2/7 x^{7/2}-8/5 x^{5/2}+8/3 x^{3/2}+C

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

\int x (x - 2)^{2} d x

\frac{2}{7} x^{\frac{7}{2}} - \frac{8}{5} x^{\frac{5}{2}} + \frac{8}{3} x^{\frac{3}{2}} + C

Solution steps

\int x (x - 2)^{2} d x

Expand

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

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

\int x^{\frac{5}{2}} d x = \frac{2}{7} x^{\frac{7}{2}}

\int 4 x^{\frac{3}{2}} d x = \frac{8}{5} x^{\frac{5}{2}}

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

= \frac{2}{7} x^{\frac{7}{2}} - \frac{8}{5} x^{\frac{5}{2}} + \frac{8}{3} x^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{7} x^{\frac{7}{2}} - \frac{8}{5} x^{\frac{5}{2}} + \frac{8}{3} x^{\frac{3}{2}} + C

What is the integral of sqrt(x)(x-2)^2 ?
The integral of sqrt(x)(x-2)^2 is 2/7 x^{7/2}-8/5 x^{5/2}+8/3 x^{3/2}+C