What is the integral of (2x^3+7x^2+2x+9)/(2x+3) ?

The integral of (2x^3+7x^2+2x+9)/(2x+3) is (x^3)/3+x^2-2x+15/2 ln|2x+3|+C

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integral of (2x^3+7x^2+2x+9)/(2x+3)

\int \frac{2 x ^{3} + 7 x ^{2} + 2 x + 9}{2 x + 3} d x

\frac{x ^{3}}{3} + x^{2} - 2 x + \frac{15}{2} ln ∣ 2 x + 3 ∣ + C

Solution steps

\int \frac{2 x ^{3} + 7 x ^{2} + 2 x + 9}{2 x + 3} d x

Long division

\frac{2 x ^{3} + 7 x ^{2} + 2 x + 9}{2 x + 3} : x^{2} + 2 x - 2 + \frac{15}{2 x + 3}

= \int x^{2} + 2 x - 2 + \frac{15}{2 x + 3} 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 x^{2} d x + \int 2 x d x - \int 2 d x + \int \frac{15}{2 x + 3} d x

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

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

\int 2 d x = 2 x

\int \frac{15}{2 x + 3} d x = \frac{15}{2} ln ∣ 2 x + 3 ∣

= \frac{x ^{3}}{3} + x^{2} - 2 x + \frac{15}{2} ln ∣ 2 x + 3 ∣

Add a constant to the solution

= \frac{x ^{3}}{3} + x^{2} - 2 x + \frac{15}{2} ln ∣ 2 x + 3 ∣ + C

What is the integral of (2x^3+7x^2+2x+9)/(2x+3) ?
The integral of (2x^3+7x^2+2x+9)/(2x+3) is (x^3)/3+x^2-2x+15/2 ln|2x+3|+C