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

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

Popular >

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

= \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

\frac{\partial}{\partial x} (2 y - sin (x))

\frac{d}{d x} (x^{3} e^{4 x})

t^{2} y^{^{''}} + 7 t y^{^{'}} + 5 y = t

\frac{d}{d x} (\frac{2}{5} x^{15})

t^{2} cos (2 t + 3 0^{\circ})