What is the integral of 3/(3+x) ?

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

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integral of 3/(3+x)

\int \frac{3}{3 + x} d x

3 ln ∣ 3 + x ∣ + C

Solution steps

\int \frac{3}{3 + x} d x

Take the constant out:

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

= 3 \cdot \int \frac{1}{3 + x} d x

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 3 ln ∣ u ∣

Substitute back

u = 3 + x

= 3 ln ∣ 3 + x ∣

Add a constant to the solution

= 3 ln ∣ 3 + x ∣ + C

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