What is the integral of 4/(8+5x) ?

The integral of 4/(8+5x) is 4/5 ln|8+5x|+C

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integral of 4/(8+5x)

\int \frac{4}{8 + 5 x} d x

\frac{4}{5} ln ∣ 8 + 5 x ∣ + C

Solution steps

\int \frac{4}{8 + 5 x} d x

Take the constant out:

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

= 4 \cdot \int \frac{1}{8 + 5 x} d x

Apply u-substitution

= 4 \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

= 4 \cdot \frac{1}{5} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 4 \cdot \frac{1}{5} ln ∣ u ∣

Substitute back

u = 8 + 5 x

= 4 \cdot \frac{1}{5} ln ∣ 8 + 5 x ∣

Simplify

4 \cdot \frac{1}{5} ln ∣ 8 + 5 x ∣ : \frac{4}{5} ln ∣ 8 + 5 x ∣

= \frac{4}{5} ln ∣ 8 + 5 x ∣

Add a constant to the solution

= \frac{4}{5} ln ∣ 8 + 5 x ∣ + C