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

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

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

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

\frac{2 x ^{5}}{5} + \frac{3 x ^{4}}{4} + 2 x^{3} + \frac{9 x ^{2}}{2} + C

Solution steps

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

Expand

(x^{3} + 3 x) (2 x + 3) : 2 x^{4} + 3 x^{3} + 6 x^{2} + 9 x

= \int 2 x^{4} + 3 x^{3} + 6 x^{2} + 9 x 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 2 x^{4} d x + \int 3 x^{3} d x + \int 6 x^{2} d x + \int 9 x d x

\int 2 x^{4} d x = \frac{2 x ^{5}}{5}

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

\int 6 x^{2} d x = 2 x^{3}

\int 9 x d x = \frac{9 x ^{2}}{2}

= \frac{2 x ^{5}}{5} + \frac{3 x ^{4}}{4} + 2 x^{3} + \frac{9 x ^{2}}{2}

Add a constant to the solution

= \frac{2 x ^{5}}{5} + \frac{3 x ^{4}}{4} + 2 x^{3} + \frac{9 x ^{2}}{2} + C

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