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Popular Calculus >

integral of x/(9+x)

Solution

\int \frac{x}{9 + x} d x

Solution

9 + x - 9 ln ∣ 9 + x ∣ + C

Solution steps

\int \frac{x}{9 + x} d x

Apply u-substitution

= \int \frac{u - 9}{u} d u

Expand

\frac{u - 9}{u} : 1 - \frac{9}{u}

= \int 1 - \frac{9}{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{9}{u} d u

\int 1 d u = u

\int \frac{9}{u} d u = 9 ln ∣ u ∣

= u - 9 ln ∣ u ∣

Substitute back

u = 9 + x

= 9 + x - 9 ln ∣ 9 + x ∣

Add a constant to the solution

= 9 + x - 9 ln ∣ 9 + x ∣ + C

Graph

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Frequently Asked Questions (FAQ)

What is the integral of x/(9+x) ?
The integral of x/(9+x) is 9+x-9ln|9+x|+C