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

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

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

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

x + \frac{3}{2} + ln ∣ 2 x + 3 ∣ + C

Solution steps

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

Expand

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

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

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

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

Add similar elements:

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

= x + \frac{3}{2} + ln ∣ 2 x + 3 ∣

Add a constant to the solution

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

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