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

\frac{x}{x - 6}

f (x) = (x^{3} + 1) e^{x}

\int_{1}^{3} (x^{2} + 2 x - 2) d x

\int x^{2} (4 x + 7) d x

\frac{d}{d x} (20 x e^{- x})

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