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

The integral of (3x^2+2x)^2 is (9x^5)/5+3x^4+(4x^3)/3+C

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

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

\frac{9 x ^{5}}{5} + 3 x^{4} + \frac{4 x ^{3}}{3} + C

Solution steps

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

Expand

(3 x^{2} + 2 x)^{2} : 9 x^{4} + 12 x^{3} + 4 x^{2}

= \int 9 x^{4} + 12 x^{3} + 4 x^{2} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 9 x^{4} d x + \int 12 x^{3} d x + \int 4 x^{2} d x

\int 9 x^{4} d x = \frac{9 x ^{5}}{5}

\int 12 x^{3} d x = 3 x^{4}

\int 4 x^{2} d x = \frac{4 x ^{3}}{3}

= \frac{9 x ^{5}}{5} + 3 x^{4} + \frac{4 x ^{3}}{3}

Add a constant to the solution

= \frac{9 x ^{5}}{5} + 3 x^{4} + \frac{4 x ^{3}}{3} + C

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What is the integral of (3x^2+2x)^2 ?
The integral of (3x^2+2x)^2 is (9x^5)/5+3x^4+(4x^3)/3+C