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

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

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integral of sqrt(x-2)x

\int x - 2 x d x

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

Solution steps

\int x - 2 x d x

Apply u-substitution

= \int (u + 2) u d u

Expand

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

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

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

\int 2 u d u = \frac{4}{3} u^{\frac{3}{2}}

= \frac{2}{5} u^{\frac{5}{2}} + \frac{4}{3} u^{\frac{3}{2}}

Substitute back

u = x - 2

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

Add a constant to the solution

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

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