What is the integral of (4x^2-7)/(2x-3) ?

The integral of (4x^2-7)/(2x-3) is x^2+3x+ln|2x-3|+C

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integral of (4x^2-7)/(2x-3)

\int \frac{4 x ^{2} - 7}{2 x - 3} d x

x^{2} + 3 x + ln ∣ 2 x - 3 ∣ + C

Solution steps

\int \frac{4 x ^{2} - 7}{2 x - 3} d x

Expand

\frac{4 x ^{2} - 7}{2 x - 3} : \frac{4 x ^{2}}{2 x - 3} - \frac{7}{2 x - 3}

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{4 x ^{2}}{2 x - 3} d x - \int \frac{7}{2 x - 3} d x

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

\int \frac{7}{2 x - 3} d x = \frac{7}{2} ln ∣ 2 x - 3 ∣

= x^{2} + 3 x + \frac{9}{2} ln ∣ 2 x - 3 ∣ - \frac{7}{2} ln ∣ 2 x - 3 ∣

Add similar elements:

\frac{9}{2} ln ∣ 2 x - 3 ∣ - \frac{7}{2} ln ∣ 2 x - 3 ∣ = ln ∣ 2 x - 3 ∣

= x^{2} + 3 x + ln ∣ 2 x - 3 ∣

Add a constant to the solution

= x^{2} + 3 x + ln ∣ 2 x - 3 ∣ + C

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What is the integral of (4x^2-7)/(2x-3) ?
The integral of (4x^2-7)/(2x-3) is x^2+3x+ln|2x-3|+C