What is the integral of 5/(8y-7) ?

The integral of 5/(8y-7) is 5/8 ln|8y-7|+C

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integral of 5/(8y-7)

\int \frac{5}{8 y - 7} d y

\frac{5}{8} ln ∣ 8 y - 7 ∣ + C

Solution steps

\int \frac{5}{8 y - 7} d y

Take the constant out:

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

= 5 \cdot \int \frac{1}{8 y - 7} d y

Apply u-substitution

= 5 \cdot \int \frac{1}{8 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}{8} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 5 \cdot \frac{1}{8} ln ∣ u ∣

Substitute back

u = 8 y - 7

= 5 \cdot \frac{1}{8} ln ∣ 8 y - 7 ∣

Simplify

5 \cdot \frac{1}{8} ln ∣ 8 y - 7 ∣ : \frac{5}{8} ln ∣ 8 y - 7 ∣

= \frac{5}{8} ln ∣ 8 y - 7 ∣

Add a constant to the solution

= \frac{5}{8} ln ∣ 8 y - 7 ∣ + C