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

The integral of (3x+5)/(2x+3) is 3/2 x+9/4+1/4 ln|2x+3|+C

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integral of (3x+5)/(2x+3)

\int \frac{3 x + 5}{2 x + 3} d x

\frac{3}{2} x + \frac{9}{4} + \frac{1}{4} ln ∣ 2 x + 3 ∣ + C

Solution steps

\int \frac{3 x + 5}{2 x + 3} d x

Expand

\frac{3 x + 5}{2 x + 3} : \frac{3 x}{2 x + 3} + \frac{5}{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{3 x}{2 x + 3} d x + \int \frac{5}{2 x + 3} d x

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

\int \frac{5}{2 x + 3} d x = \frac{5}{2} ln ∣ 2 x + 3 ∣

= \frac{3}{4} (2 x + 3 - 3 ln ∣ 2 x + 3 ∣) + \frac{5}{2} ln ∣ 2 x + 3 ∣

Simplify

\frac{3}{4} (2 x + 3 - 3 ln ∣ 2 x + 3 ∣) + \frac{5}{2} ln ∣ 2 x + 3 ∣ : \frac{3}{2} x + \frac{9}{4} + \frac{1}{4} ln ∣ 2 x + 3 ∣

= \frac{3}{2} x + \frac{9}{4} + \frac{1}{4} ln ∣ 2 x + 3 ∣

Add a constant to the solution

= \frac{3}{2} x + \frac{9}{4} + \frac{1}{4} ln ∣ 2 x + 3 ∣ + C

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What is the integral of (3x+5)/(2x+3) ?
The integral of (3x+5)/(2x+3) is 3/2 x+9/4+1/4 ln|2x+3|+C