Question

Evaluate \[\int\frac{dx}{3x^2+13x-10}\].

Options

• \[\frac{1}{6}\log\left|\frac{3x-2}{x+5}\right|+C\]

• \[\frac{1}{17}\log\left|\frac{3x+2}{x-5}\right|+C\]

• \[\frac{1}{17}\tan^{-1}\left(\frac{3x-2}{x+5}\right)+C\]

• \[\frac{1}{17}\log\left|\frac{3x-2}{x+5}\right|+C\]

Answer 1

Complete the square:

\[3x^2 + 13x - 10 = 3\left[\left(x + \frac{13}{6}\right)^2 - \left(\frac{17}{6}\right)^2\right]. \]

Put \[t = x + \frac{13}{6}.\] Using \[\int \frac{dt}{t^2 - a^2} = \frac{1}{2a}\log\left|\frac{t-a}{t+a}\right| + C,\]

we get \[ \frac{1}{17}\log\left|\frac{3x-2}{x+5}\right| + C.\]