What is the integral of (x/(x-5)) ?

The integral of (x/(x-5)) is x-5+5ln|x-5|+C

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integral of (x/(x-5))

\int (\frac{x}{x - 5}) d x

x - 5 + 5 ln ∣ x - 5 ∣ + C

Solution steps

\int \frac{x}{x - 5} d x

Apply u-substitution

= \int \frac{u + 5}{u} d u

Expand

\frac{u + 5}{u} : 1 + \frac{5}{u}

= \int 1 + \frac{5}{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{5}{u} d u

\int 1 d u = u

\int \frac{5}{u} d u = 5 ln ∣ u ∣

= u + 5 ln ∣ u ∣

Substitute back

u = x - 5

= x - 5 + 5 ln ∣ x - 5 ∣

Add a constant to the solution

= x - 5 + 5 ln ∣ x - 5 ∣ + C