\(\int \frac{x}{2x^2-x-1}dx\)
\(\int \frac{x}{2x^2-2x+x-1}dx\)
\(\int \frac{x}{2x(x-1)+1(x-1)}dx\)
\(\int \frac{x}{(2x+1)(x-1)}dx\)
Using partial fraction,
\( \frac{x}{(2x+1)(x-1)}= \frac{A}{2x+1}+\frac{B}{x-1}\)
\(\frac{x}{(2x+1)(x-1)}= \frac{A(x-1)+B(2x+1)}{(2x+1)(x-1)}\)
\(\frac{x}{(2x+1)(x-1)}= \frac{Ax-A+2xB+B}{(2x+1)(x-1)}\)
\(\frac{x}{(2x+1)(x-1)}= \frac{x(A+2B)+(B-A)}{(2x+1)(x-1)}\)
Equating like terms,
\(A+2B=1\)
\(B-A=0\)
After solving,, we will
\(A=B=\frac{1}{3}\)
\(\frac{x}{(2x+1)(x-1)}= \frac{1}{3(2x+1)}+\frac{1}{3(x-1)}\)
\(\int (\frac{1}{3(2x+1)}+\frac{1}{3(x-1)})dx\)
\(\int \frac{1}{3(2x+1)}dx+\frac{1}{3(x-1)}dx\)
\(\frac{1}{3} \frac{1}{2} ln|2x+1|+\frac{1}{3}ln|x-1|+c\)
\(\frac{1}{3} (\frac{1}{2} ln|2x+1|+ln|x-1|)+c\)
