What is the integral of 9(x^2+3x+5)(2x+3) ?

The integral of 9(x^2+3x+5)(2x+3) is 9((x^4)/2+3x^3+(19x^2)/2+15x)+C

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integral of 9(x^2+3x+5)(2x+3)

\int 9 (x^{2} + 3 x + 5) (2 x + 3) d x

9 (\frac{x ^{4}}{2} + 3 x^{3} + \frac{19 x ^{2}}{2} + 15 x) + C

Solution steps

\int 9 (x^{2} + 3 x + 5) (2 x + 3) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 9 \cdot \int (x^{2} + 3 x + 5) (2 x + 3) d x

Expand

(x^{2} + 3 x + 5) (2 x + 3) : 2 x^{3} + 9 x^{2} + 19 x + 15

= 9 \cdot \int 2 x^{3} + 9 x^{2} + 19 x + 15 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

= 9 (\int 2 x^{3} d x + \int 9 x^{2} d x + \int 19 x d x + \int 15 d x)

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

\int 9 x^{2} d x = 3 x^{3}

\int 19 x d x = \frac{19 x ^{2}}{2}

\int 15 d x = 15 x

= 9 (\frac{x ^{4}}{2} + 3 x^{3} + \frac{19 x ^{2}}{2} + 15 x)

Add a constant to the solution

= 9 (\frac{x ^{4}}{2} + 3 x^{3} + \frac{19 x ^{2}}{2} + 15 x) + C

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What is the integral of 9(x^2+3x+5)(2x+3) ?
The integral of 9(x^2+3x+5)(2x+3) is 9((x^4)/2+3x^3+(19x^2)/2+15x)+C