What is the integral of x/((x-6)) ?

The integral of x/((x-6)) is x-6+6ln|x-6|+C

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integral of x/((x-6))

\int \frac{x}{( x - 6 )} d x

x - 6 + 6 ln ∣ x - 6 ∣ + C

Solution steps

\int \frac{x}{x - 6} d x

Apply u-substitution

= \int \frac{u + 6}{u} d u

Expand

\frac{u + 6}{u} : 1 + \frac{6}{u}

= \int 1 + \frac{6}{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{6}{u} d u

\int 1 d u = u

\int \frac{6}{u} d u = 6 ln ∣ u ∣

= u + 6 ln ∣ u ∣

Substitute back

u = x - 6

= x - 6 + 6 ln ∣ x - 6 ∣

Add a constant to the solution

= x - 6 + 6 ln ∣ x - 6 ∣ + C