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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

Add similar elements:

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

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

Add a constant to the solution

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

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