What is the integral of x(sqrt(x-3)) ?

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

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integral of x(sqrt(x-3))

\int x (x - 3) d x

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

Solution steps

\int x x - 3 d x

Apply u-substitution

: \int (u + 3) u d u

= \int (u + 3) u d u

Expand

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

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

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

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

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

Substitute back

u = x - 3

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

Add a constant to the solution

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

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