What is the integral of sqrt(x^3+3x^2) ?

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

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integral of sqrt(x^3+3x^2)

\int x^{3} + 3 x^{2} d x

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

Solution steps

\int x^{3} + 3 x^{2} d x

Simplify

x^{3} + 3 x^{2} : x x + 3

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