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Popular Calculus >

integral of x/(3+x)

Solution

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

Solution

3 + x - 3 ln ∣ 3 + x ∣ + C

Solution steps

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

Apply u-substitution

= \int \frac{u - 3}{u} d u

Expand

\frac{u - 3}{u} : 1 - \frac{3}{u}

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int 1 d u = u

\int \frac{3}{u} d u = 3 ln ∣ u ∣

= u - 3 ln ∣ u ∣

Substitute back

u = 3 + x

= 3 + x - 3 ln ∣ 3 + x ∣

Add a constant to the solution

= 3 + x - 3 ln ∣ 3 + x ∣ + C

Graph

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Frequently Asked Questions (FAQ)

What is the integral of x/(3+x) ?
The integral of x/(3+x) is 3+x-3ln|3+x|+C