Question

Evaluate:
$\int \frac{x-x^2}{x^2-2x-3}dx$

Answer 1

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

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

$\frac{x-x^2}{(x-3)(x+1)}=\frac{A}{x-3}+\frac{B}{x+1}$

$x-x^2=A(x+1)+(x-3)$

put $x=-1,$ $\Rightarrow -1-1=0+B(-1-3)$

$\Rightarrow$ $-2=-4B$ or $B=\frac{1}{2}$

put $x=3$ $\Rightarrow 3-9=A(3+1)+0$

$\Rightarrow$ $-6=4A$ or $A=\frac{-6}{4}=\frac{-3}{2}$

$\therefore \int \frac{x-x^{2}dx}{x^{2}2x-3}=\frac{-3}{2}\int \frac{dx}{x-3}+\frac{1}{2}\int \frac{dx}{x+1}$

$=\frac{-3}{2}log(x-3)+\frac{1}{2}log(x+1)+C$