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

The integral of (x^2)/(2x-3) is (x^2)/4+3/4 x+9/8 ln|2x-3|+C

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integral of (x^2)/(2x-3)

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

\frac{x ^{2}}{4} + \frac{3}{4} x + \frac{9}{8} ln ∣ 2 x - 3 ∣ + C

Solution steps

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

Long division

\frac{x ^{2}}{2 x - 3} : \frac{x}{2} + \frac{3}{4} + \frac{9}{4 ( 2 x - 3 )}

= \int \frac{x}{2} + \frac{3}{4} + \frac{9}{4 ( 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 \frac{x}{2} d x + \int \frac{3}{4} d x + \int \frac{9}{4 ( 2 x - 3 )} d x

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

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

\int \frac{9}{4 ( 2 x - 3 )} d x = \frac{9}{8} ln ∣ 2 x - 3 ∣

= \frac{x ^{2}}{4} + \frac{3}{4} x + \frac{9}{8} ln ∣ 2 x - 3 ∣

Add a constant to the solution

= \frac{x ^{2}}{4} + \frac{3}{4} x + \frac{9}{8} ln ∣ 2 x - 3 ∣ + C

What is the integral of (x^2)/(2x-3) ?
The integral of (x^2)/(2x-3) is (x^2)/4+3/4 x+9/8 ln|2x-3|+C