Question

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

Answer 1

Now,

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

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

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

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

$=\frac{1}{3}\log \left|\frac{x-1}{x+2}\right|$$+\frac{1}{2}\log \left|\frac{x-3}{x-1}\right|+c$ [ Where $c$ is integrating constant]