What is the integral of x/(x-8) ?

The integral of x/(x-8) is x-8+8ln|x-8|+C

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integral of x/(x-8)

\int \frac{x}{x - 8} d x

x - 8 + 8 ln ∣ x - 8 ∣ + C

Solution steps

\int \frac{x}{x - 8} d x

Apply u-substitution

: \int \frac{u + 8}{u} d u

= \int \frac{u + 8}{u} d u

Expand

\frac{u + 8}{u} : 1 + \frac{8}{u}

= \int 1 + \frac{8}{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{8}{u} d u

\int 1 d u = u

\int \frac{8}{u} d u = 8 ln ∣ u ∣

= u + 8 ln ∣ u ∣

Substitute back

u = x - 8

= x - 8 + 8 ln ∣ x - 8 ∣

Add a constant to the solution

= x - 8 + 8 ln ∣ x - 8 ∣ + C

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