What is the integral of 2/(y+3) ?

The integral of 2/(y+3) is 2ln|y+3|+C

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integral of 2/(y+3)

\int \frac{2}{y + 3} d y

2 ln ∣ y + 3 ∣ + C

Solution steps

\int \frac{2}{y + 3} d y

Take the constant out:

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

= 2 \cdot \int \frac{1}{y + 3} d y

Apply u-substitution

= 2 \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 ln ∣ u ∣

Substitute back

u = y + 3

= 2 ln ∣ y + 3 ∣

Add a constant to the solution

= 2 ln ∣ y + 3 ∣ + C