What is the integral of (19)/(2+5y) ?

The integral of (19)/(2+5y) is 19/5 ln|2+5y|+C

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integral of (19)/(2+5y)

\int \frac{19}{2 + 5 y} d y

\frac{19}{5} ln ∣ 2 + 5 y ∣ + C

Solution steps

\int \frac{19}{2 + 5 y} d y

Take the constant out:

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

= 19 \cdot \int \frac{1}{2 + 5 y} d y

Apply u-substitution

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

= 19 \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

= 19 \cdot \frac{1}{5} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 19 \cdot \frac{1}{5} ln ∣ u ∣

Substitute back

u = 2 + 5 y

= 19 \cdot \frac{1}{5} ln ∣ 2 + 5 y ∣

Simplify

19 \cdot \frac{1}{5} ln ∣ 2 + 5 y ∣ : \frac{19}{5} ln ∣ 2 + 5 y ∣

= \frac{19}{5} ln ∣ 2 + 5 y ∣

Add a constant to the solution

= \frac{19}{5} ln ∣ 2 + 5 y ∣ + C

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What is the integral of (19)/(2+5y) ?
The integral of (19)/(2+5y) is 19/5 ln|2+5y|+C