Question

Integrate :-
${\int }^{}\frac{1}{x-x^3}dx$

Answer 1

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

$\frac{dx}{x(1+x)(1-x)}=A\int \frac{dx}{x}+B\int \frac{dx}{1+x}+C\int \frac{dx}{1-x}$

$1=A(1+x)(1-x)+Bx(1-x)+Cx(1+x)$

Put $x=0$

$A=1$

Put $x=1$

$C=\frac{1}{2}$

Put $x=-1$
$B=\frac{1}{2}$

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

$=ln|x|+\frac{1}{2}ln|1+x|-\frac{1}{2}ln|1-x|+C$