Question

Integrate:

$\int \frac{ax+b}{cx+d}dx$

Answer 1

$\frac{ax+b}{cx+d}==\frac{ax}{cx+d}+\frac{b}{cx+d}$

Using sum rule,$\int \frac{ax}{cx+d}dx+\int \frac{b}{cx+d}dx$

$\int \frac{ax}{cx+d}dx=a\frac{1}{c^2}(cx+d-d\ln |cx+d|)$

$\int \frac{b}{cx+d}dx=b\frac{1}{c}\ln |cx+d|$

$=a\frac{1}{c^2}(cx+d-d\ln |cx+d|)+b\frac{1}{c}\ln |cx+d|$

Adding constant$=a\frac{1}{c^2}(cx+d-d\ln |cx+d|)+b\frac{1}{c}\ln |cx+d|+C$