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

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

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integral of 3/(2x-3)

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

\frac{3}{2} ln ∣ 2 x - 3 ∣ + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int \frac{1}{2 x - 3} d x

Apply u-substitution

= 3 \cdot \int \frac{1}{2 u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \frac{1}{2} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 3 \cdot \frac{1}{2} ln ∣ u ∣

Substitute back

u = 2 x - 3

= 3 \cdot \frac{1}{2} ln ∣ 2 x - 3 ∣

Simplify

3 \cdot \frac{1}{2} ln ∣ 2 x - 3 ∣ : \frac{3}{2} ln ∣ 2 x - 3 ∣

= \frac{3}{2} ln ∣ 2 x - 3 ∣

Add a constant to the solution

= \frac{3}{2} ln ∣ 2 x - 3 ∣ + C