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

The integral of (2x^3+2)(x^2+3) is (x^6)/3+(3x^4)/2+(2x^3)/3+6x+C

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

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

\frac{x ^{6}}{3} + \frac{3 x ^{4}}{2} + \frac{2 x ^{3}}{3} + 6 x + C

Solution steps

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

Expand

(2 x^{3} + 2) (x^{2} + 3) : 2 x^{5} + 6 x^{3} + 2 x^{2} + 6

= \int 2 x^{5} + 6 x^{3} + 2 x^{2} + 6 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^{5} d x + \int 6 x^{3} d x + \int 2 x^{2} d x + \int 6 d x

\int 2 x^{5} d x = \frac{x ^{6}}{3}

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

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

\int 6 d x = 6 x

= \frac{x ^{6}}{3} + \frac{3 x ^{4}}{2} + \frac{2 x ^{3}}{3} + 6 x

Add a constant to the solution

= \frac{x ^{6}}{3} + \frac{3 x ^{4}}{2} + \frac{2 x ^{3}}{3} + 6 x + C

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What is the integral of (2x^3+2)(x^2+3) ?
The integral of (2x^3+2)(x^2+3) is (x^6)/3+(3x^4)/2+(2x^3)/3+6x+C