What is the integral of 3/(5y+4) ?

The integral of 3/(5y+4) is 3/5 ln|5y+4|+C

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integral of 3/(5y+4)

\int \frac{3}{5 y + 4} d y

\frac{3}{5} ln ∣ 5 y + 4 ∣ + C

Solution steps

\int \frac{3}{5 y + 4} d y

Take the constant out:

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

= 3 \cdot \int \frac{1}{5 y + 4} d y

Apply u-substitution

: \int \frac{1}{5 u} d u

= 3 \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

= 3 \cdot \frac{1}{5} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 3 \cdot \frac{1}{5} ln ∣ u ∣

Substitute back

u = 5 y + 4

= 3 \cdot \frac{1}{5} ln ∣ 5 y + 4 ∣

Simplify

3 \cdot \frac{1}{5} ln ∣ 5 y + 4 ∣ : \frac{3}{5} ln ∣ 5 y + 4 ∣

= \frac{3}{5} ln ∣ 5 y + 4 ∣

Add a constant to the solution

= \frac{3}{5} ln ∣ 5 y + 4 ∣ + C

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