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

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

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integral of (4x+3)/(2x^2+3x)

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

ln ∣ 2 x + 3 ∣ + ln ∣ x ∣ + C

Solution steps

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

Expand

\frac{4 x + 3}{2 x ^{2} + 3 x} : \frac{4 x}{2 x ^{2} + 3 x} + \frac{3}{2 x ^{2} + 3 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{4 x}{2 x ^{2} + 3 x} d x + \int \frac{3}{2 x ^{2} + 3 x} d x

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

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

= 2 ln ∣ 2 x + 3 ∣ + ln ∣ x ∣ - ln ∣ 2 x + 3 ∣

Simplify

= ln ∣ 2 x + 3 ∣ + ln ∣ x ∣

Add a constant to the solution

= ln ∣ 2 x + 3 ∣ + ln ∣ x ∣ + C

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