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

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

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

integral

\frac{2}{( 2 x + 3 )}

ln ∣ 2 x + 3 ∣ + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

= 2 \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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 \cdot \frac{1}{2} ln ∣ u ∣

Substitute back

u = 2 x + 3

= 2 \cdot \frac{1}{2} ln ∣ 2 x + 3 ∣

Simplify

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

= ln ∣ 2 x + 3 ∣

Add a constant to the solution

= ln ∣ 2 x + 3 ∣ + C