What is the integral of (3-5x+7x^2)/(3x^3-19x^2-14x) ?

The integral of (3-5x+7x^2)/(3x^3-19x^2-14x) is -3/14 ln|x|+85/138 ln|3x+2|+311/161 ln|x-7|+C

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integral of (3-5x+7x^2)/(3x^3-19x^2-14x)

\int \frac{3 - 5 x + 7 x ^{2}}{3 x ^{3} - 19 x ^{2} - 14 x} d x

- \frac{3}{14} ln ∣ x ∣ + \frac{85}{138} ln ∣ 3 x + 2 ∣ + \frac{311}{161} ln ∣ x - 7 ∣ + C

Solution steps

\int \frac{3 - 5 x + 7 x ^{2}}{3 x ^{3} - 19 x ^{2} - 14 x} d x

Take the partial fraction of

\frac{3 - 5 x + 7 x ^{2}}{3 x ^{3} - 19 x ^{2} - 14 x} : - \frac{3}{14 x} + \frac{85}{46 ( 3 x + 2 )} + \frac{311}{161 ( x - 7 )}

= \int - \frac{3}{14 x} + \frac{85}{46 ( 3 x + 2 )} + \frac{311}{161 ( x - 7 )} d x

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{3}{14 x} d x + \int \frac{85}{46 ( 3 x + 2 )} d x + \int \frac{311}{161 ( x - 7 )} d x

\int \frac{3}{14 x} d x = \frac{3}{14} ln ∣ x ∣

\int \frac{85}{46 ( 3 x + 2 )} d x = \frac{85}{138} ln ∣ 3 x + 2 ∣

\int \frac{311}{161 ( x - 7 )} d x = \frac{311}{161} ln ∣ x - 7 ∣

= - \frac{3}{14} ln ∣ x ∣ + \frac{85}{138} ln ∣ 3 x + 2 ∣ + \frac{311}{161} ln ∣ x - 7 ∣

Add a constant to the solution

= - \frac{3}{14} ln ∣ x ∣ + \frac{85}{138} ln ∣ 3 x + 2 ∣ + \frac{311}{161} ln ∣ x - 7 ∣ + C

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What is the integral of (3-5x+7x^2)/(3x^3-19x^2-14x) ?
The integral of (3-5x+7x^2)/(3x^3-19x^2-14x) is -3/14 ln|x|+85/138 ln|3x+2|+311/161 ln|x-7|+C