What is the integral of 4/(5+6x) ?

The integral of 4/(5+6x) is 2/3 ln|5+6x|+C

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integral of 4/(5+6x)

\int \frac{4}{5 + 6 x} d x

\frac{2}{3} ln ∣ 5 + 6 x ∣ + C

Solution steps

\int \frac{4}{5 + 6 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}{5 + 6 x} d x

Apply u-substitution

: \int \frac{1}{6 u} d u

= 4 \cdot \int \frac{1}{6 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}{6} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 4 \cdot \frac{1}{6} ln ∣ u ∣

Substitute back

u = 5 + 6 x

= 4 \cdot \frac{1}{6} ln ∣ 5 + 6 x ∣

Simplify

4 \cdot \frac{1}{6} ln ∣ 5 + 6 x ∣ : \frac{2}{3} ln ∣ 5 + 6 x ∣

= \frac{2}{3} ln ∣ 5 + 6 x ∣

Add a constant to the solution

= \frac{2}{3} ln ∣ 5 + 6 x ∣ + C

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