Question

Find $\int \frac{2x-1}{(x+1)(x+3)}dx$

Answer 1

$\int \frac{2x-1}{(x+1)(x+3)}dx$

$=\int [\frac{2x}{(x+1)(x+3)}-\frac{1}{(x+1)(x+3)}]dx$

$=\int [\frac{3}{x+3}-\frac{1}{x+1}-\frac{1}{2}(\frac{1}{x+1}-\frac{1}{x+3})]dx$

$=3\ln (x+3)-\ln (x+1)-\frac{1}{2}\ln (x+1)+\frac{1}{2}\ln (x+3)=\frac{7}{2}\ln (x+3)-\frac{3}{2}\ln (x+1)$