`int1/(3x^2+13x-10)dx`

Explanation

`int frac{1}{3 x^{2}+13 x-10} d x `

`= frac{1}{3} int frac{1}{x^{2}+frac{13}{3} x-frac{10}{3}} d x `

`= frac{1}{3} int frac{1}{x^{2}+frac{13 x}{3}+(frac{13}{6})^{2}-(frac{13}{6})^{2}-frac{10}{3}} d x `

`= frac{1}{3} int frac{1}{(x+frac{13}{6})^{2}-frac{169}{36}-frac{10}{3}} d x `

`= frac{1}{3} int frac{1}{(x+frac{13}{6})^{2}-frac{169-120}{36}} d x `

`= frac{1}{3} int frac{1}{(x+frac{13}{6})^{2}-(frac{17}{6})^{2}} d x `

`= frac{1}{3} times frac{1}{2 times frac{17}{6}} ln |frac{x+frac{13}{6}-frac{17}{6}}{x+frac{13}{6}+frac{17}{6}}| cdots cdots cdots cdots[because frac{1}{x^{2}-a^{2}} d x=frac{1}{2 a} ln |frac{x-a}{x+a}|+C] `

`= frac{1}{17} ln |frac{x-frac{2}{3}}{x+5}|+C `

...