How do I find the integral intx/(x-6)dx ?

Sep 30, 2014

By Substitution,

$\int \frac{x}{x - 6} \mathrm{dx} = x + 6 \ln | x - 6 | + C$

Let us look at some details.

$\int \frac{x}{x - 6} \mathrm{dx}$

by the sunstitution $u = x - 6$,
$R i g h t a r r o w x = u + 6$
$R i g h t a r r o w \mathrm{dx} = \mathrm{du}$

$= \int \frac{u + 6}{u} \mathrm{du}$

by splitting the integrand,

$= \int \left(1 + \frac{6}{u}\right) \mathrm{du}$

$= u + 6 \ln | u | + {C}_{1}$

by putting $u = x - 6$ back in,

$= x - 6 + 6 \ln | x - 6 | + {C}_{1}$

by letting $C = {C}_{1} - 6$,

$= x + 6 \ln | x - 6 | + C$

I hope that this was helpful.