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

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

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integral of 5/(5x+3)

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

ln ∣ 5 x + 3 ∣ + C

Solution steps

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

Take the constant out:

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

= 5 \cdot \int \frac{1}{5 x + 3} d x

Apply u-substitution

= 5 \cdot \int \frac{1}{5 u} d u

Take the constant out:

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

= 5 \cdot \frac{1}{5} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 5 \cdot \frac{1}{5} ln ∣ u ∣

Substitute back

u = 5 x + 3

= 5 \cdot \frac{1}{5} ln ∣ 5 x + 3 ∣

Simplify

5 \cdot \frac{1}{5} ln ∣ 5 x + 3 ∣ : ln ∣ 5 x + 3 ∣

= ln ∣ 5 x + 3 ∣

Add a constant to the solution

= ln ∣ 5 x + 3 ∣ + C