Question

$\int \frac{dx}{3x^2+13x-10}$

Answer 1

$\int \frac{dx}{3x^2+13x-10}$

$=\frac{1}{3}\int \frac{dx}{x^2+\frac{13}{3}x-\frac{10}{3}}$

$=\frac{1}{3}\int \frac{dx}{x^2+\frac{13}{3}x+\frac{169}{36}-\frac{169}{36}-\frac{10}{3}}$

$=\frac{1}{3}\int \frac{dx}{{\left(x^2+\frac{13}{6}\right)}^2-{\left(\frac{17}{6}\right)}^2}$

$=\frac{1}{3}\times \frac{1}{2\times \frac{17}{6}}\log \left|\frac{x+\frac{13}{6}-\frac{17}{6}}{x+\frac{13}{6}+\frac{17}{6}}\right|+c$

$=\frac{1}{17}\log \left|\frac{6x-4}{6x+30}\right|+c$

$=\frac{1}{17}\log \left|\frac{3x-2}{3x+15}\right|c$