What is the integral of (6t-7)/(3t^2-7t+15) ?

The integral of (6t-7)/(3t^2-7t+15) is ln|36t^2-84t+180|+C

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integral of (6t-7)/(3t^2-7t+15)

\int \frac{6 t - 7}{3 t ^{2} - 7 t + 15} d t

ln 36 t^{2} - 84 t + 180 + C

Solution steps

\int \frac{6 t - 7}{3 t ^{2} - 7 t + 15} d t

Expand

\frac{6 t - 7}{3 t ^{2} - 7 t + 15} : \frac{6 t}{3 t ^{2} - 7 t + 15} - \frac{7}{3 t ^{2} - 7 t + 15}

Apply the Sum Rule:

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

= \int \frac{6 t}{3 t ^{2} - 7 t + 15} d t - \int \frac{7}{3 t ^{2} - 7 t + 15} d t

\int \frac{6 t}{3 t ^{2} - 7 t + 15} d t = ln 36 t^{2} - 84 t + 180 + \frac{14}{131} arctan (\frac{6 t - 7}{131})

\int \frac{7}{3 t ^{2} - 7 t + 15} d t = \frac{14}{131} arctan (\frac{6 t - 7}{131})

= ln 36 t^{2} - 84 t + 180 + \frac{14}{131} arctan (\frac{6 t - 7}{131}) - \frac{14}{131} arctan (\frac{6 t - 7}{131})

Add similar elements:

\frac{14}{131} arctan (\frac{6 t - 7}{131}) - \frac{14}{131} arctan (\frac{6 t - 7}{131}) = 0

= ln 36 t^{2} - 84 t + 180

Add a constant to the solution

= ln 36 t^{2} - 84 t + 180 + C

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What is the integral of (6t-7)/(3t^2-7t+15) ?
The integral of (6t-7)/(3t^2-7t+15) is ln|36t^2-84t+180|+C