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

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

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

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

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

Solution steps

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

Apply u-substitution

= \int u^{2} d u

Apply the Power Rule

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

Substitute back

u = 2 x + x^{2}

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

Simplify

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

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

Add a constant to the solution

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

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