What is the integral of (2x+1)sqrt(x^2+x) ?

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

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integral of (2x+1)sqrt(x^2+x)

\int (2 x + 1) x^{2} + x d x

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

Solution steps

\int (2 x + 1) x^{2} + x d x

Apply u-substitution

= \int u d u

Apply the Power Rule

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

Substitute back

u = x^{2} + x

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

Add a constant to the solution

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

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