What is the integral of (9t^2-26t-5)/(3t^2-5t-2) ?

The integral of (9t^2-26t-5)/(3t^2-5t-2) is

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integral of (9t^2-26t-5)/(3t^2-5t-2)

\int \frac{9 t ^{2} - 26 t - 5}{3 t ^{2} - 5 t - 2} d t

3 t - 22 (\frac{1}{12} ln 36 t^{2} - 60 t - 24 - \frac{5}{84} (ln \frac{2}{7} + \frac{6}{7} t - ln - \frac{12}{7} + \frac{6}{7} t)) - \frac{1}{7} (ln \frac{2}{7} + \frac{6}{7} t - ln - \frac{12}{7} + \frac{6}{7} t) + C

Solution steps

\int \frac{9 t ^{2} - 26 t - 5}{3 t ^{2} - 5 t - 2} d t

Long division

\frac{9 t ^{2} - 26 t - 5}{3 t ^{2} - 5 t - 2} : 3 + \frac{- 11 t + 1}{3 t ^{2} - 5 t - 2}

= \int 3 + \frac{- 11 t + 1}{3 t ^{2} - 5 t - 2} d t

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 3 d t + \int \frac{- 11 t + 1}{3 t ^{2} - 5 t - 2} d t

\int 3 d t = 3 t

\int \frac{- 11 t + 1}{3 t ^{2} - 5 t - 2} d t = - 22 (\frac{1}{12} ln 36 t^{2} - 60 t - 24 - \frac{5}{84} (ln \frac{2}{7} + \frac{6}{7} t - ln - \frac{12}{7} + \frac{6}{7} t)) - \frac{1}{7} (ln \frac{2}{7} + \frac{6}{7} t - ln - \frac{12}{7} + \frac{6}{7} t)

= 3 t - 22 (\frac{1}{12} ln 36 t^{2} - 60 t - 24 - \frac{5}{84} (ln \frac{2}{7} + \frac{6}{7} t - ln - \frac{12}{7} + \frac{6}{7} t)) - \frac{1}{7} (ln \frac{2}{7} + \frac{6}{7} t - ln - \frac{12}{7} + \frac{6}{7} t)

Add a constant to the solution

= 3 t - 22 (\frac{1}{12} ln 36 t^{2} - 60 t - 24 - \frac{5}{84} (ln \frac{2}{7} + \frac{6}{7} t - ln - \frac{12}{7} + \frac{6}{7} t)) - \frac{1}{7} (ln \frac{2}{7} + \frac{6}{7} t - ln - \frac{12}{7} + \frac{6}{7} t) + C

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What is the integral of (9t^2-26t-5)/(3t^2-5t-2) ?
The integral of (9t^2-26t-5)/(3t^2-5t-2) is