What is the integral of (6x-7)/(3x^2-7x+11) ?

The integral of (6x-7)/(3x^2-7x+11) is ln|36x^2-84x+132|+C

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integral of (6x-7)/(3x^2-7x+11)

\int \frac{6 x - 7}{3 x ^{2} - 7 x + 11} d x

ln 36 x^{2} - 84 x + 132 + C

Solution steps

\int \frac{6 x - 7}{3 x ^{2} - 7 x + 11} d x

Expand

\frac{6 x - 7}{3 x ^{2} - 7 x + 11} : \frac{6 x}{3 x ^{2} - 7 x + 11} - \frac{7}{3 x ^{2} - 7 x + 11}

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 + 11} d x - \int \frac{7}{3 x ^{2} - 7 x + 11} d x

\int \frac{6 x}{3 x ^{2} - 7 x + 11} d x = ln 36 x^{2} - 84 x + 132 + \frac{14}{83} arctan (\frac{6 x - 7}{83})

\int \frac{7}{3 x ^{2} - 7 x + 11} d x = \frac{14}{83} arctan (\frac{6 x - 7}{83})

= ln 36 x^{2} - 84 x + 132 + \frac{14}{83} arctan (\frac{6 x - 7}{83}) - \frac{14}{83} arctan (\frac{6 x - 7}{83})

Add similar elements:

\frac{14}{83} arctan (\frac{6 x - 7}{83}) - \frac{14}{83} arctan (\frac{6 x - 7}{83}) = 0

= ln 36 x^{2} - 84 x + 132

Add a constant to the solution

= ln 36 x^{2} - 84 x + 132 + C

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What is the integral of (6x-7)/(3x^2-7x+11) ?
The integral of (6x-7)/(3x^2-7x+11) is ln|36x^2-84x+132|+C