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

The integral of (3x^2+2x)sqrt(x^3+x^2) is 2/3 x^3(x+1)^{3/2}+C

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

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

\frac{2}{3} x^{3} (x + 1)^{\frac{3}{2}} + C

Solution steps

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

Apply u-substitution

= \int u d u

Apply the Power Rule

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

Substitute back

u = x^{3} + x^{2}

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

(x^{3} + x^{2})^{\frac{3}{2}} = x^{3} (x + 1)^{\frac{3}{2}}

= \frac{2}{3} x^{3} (x + 1)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{3} x^{3} (x + 1)^{\frac{3}{2}} + C

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What is the integral of (3x^2+2x)sqrt(x^3+x^2) ?
The integral of (3x^2+2x)sqrt(x^3+x^2) is 2/3 x^3(x+1)^{3/2}+C