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

The integral of (7x-3)/(x^3+2x^2-3x) is ln|x|+ln|x-1|-2ln|x+3|+C

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

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

ln ∣ x ∣ + ln ∣ x - 1 ∣ - 2 ln ∣ x + 3 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{7 x - 3}{x ^{3} + 2 x ^{2} - 3 x} : \frac{1}{x} + \frac{1}{x - 1} - \frac{2}{x + 3}

= \int \frac{1}{x} + \frac{1}{x - 1} - \frac{2}{x + 3} 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{1}{x} d x + \int \frac{1}{x - 1} d x - \int \frac{2}{x + 3} d x

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{1}{x - 1} d x = ln ∣ x - 1 ∣

\int \frac{2}{x + 3} d x = 2 ln ∣ x + 3 ∣

= ln ∣ x ∣ + ln ∣ x - 1 ∣ - 2 ln ∣ x + 3 ∣

Add a constant to the solution

= ln ∣ x ∣ + ln ∣ x - 1 ∣ - 2 ln ∣ x + 3 ∣ + C

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What is the integral of (7x-3)/(x^3+2x^2-3x) ?
The integral of (7x-3)/(x^3+2x^2-3x) is ln|x|+ln|x-1|-2ln|x+3|+C