Question

Evaluate :
$I=\int \frac{2x}{x^2-6x+6}dx$

Answer 1

$\int \frac{2x}{x^2-6x+6}dx$

put $x^2-6x+6=t$

$(2x-6)dx=dt$

$\int \frac{2x}{x^2-6x+6}dx$

$=\int \frac{2x-6}{x^2-6x+6}dx+\int \frac{6}{x^2-6x+6}dx$

$=\int \frac{dt}{t}+\int \frac{6}{(x-3{)}^2-(\sqrt{3}{)}^2}dx$

$=\log (x^2-6x+6)+6\cdot \frac{1}{2\cdot \sqrt{3}}\log \left(\frac{x-3-\sqrt{3}}{x-3+\sqrt{3}}\right)+c$

$=\log (x^2-6x+6)+\sqrt{3}\log \left(\frac{x-3-\sqrt{3}}{x-3+\sqrt{3}}\right)+c$