What is the integral of (2x^2+x-3)/(x^3+2x^2-7x+4) ?

The integral of (2x^2+x-3)/(x^3+2x^2-7x+4) is ln|x-1|+ln|x+4|+C

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integral of (2x^2+x-3)/(x^3+2x^2-7x+4)

\int \frac{2 x ^{2} + x - 3}{x ^{3} + 2 x ^{2} - 7 x + 4} d x

ln ∣ x - 1 ∣ + ln ∣ x + 4 ∣ + C

Solution steps

\int \frac{2 x ^{2} + x - 3}{x ^{3} + 2 x ^{2} - 7 x + 4} d x

Simplify

\frac{2 x ^{2} + x - 3}{x ^{3} + 2 x ^{2} - 7 x + 4} : \frac{2 x + 3}{( x - 1 ) ( x + 4 )}

= \int \frac{2 x + 3}{( x - 1 ) ( x + 4 )} d x

Take the partial fraction of

\frac{2 x + 3}{( x - 1 ) ( x + 4 )} : \frac{1}{x - 1} + \frac{1}{x + 4}

= \int \frac{1}{x - 1} + \frac{1}{x + 4} 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 - 1} d x + \int \frac{1}{x + 4} d x

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

\int \frac{1}{x + 4} d x = ln ∣ x + 4 ∣

= ln ∣ x - 1 ∣ + ln ∣ x + 4 ∣

Add a constant to the solution

= ln ∣ x - 1 ∣ + ln ∣ x + 4 ∣ + C

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What is the integral of (2x^2+x-3)/(x^3+2x^2-7x+4) ?
The integral of (2x^2+x-3)/(x^3+2x^2-7x+4) is ln|x-1|+ln|x+4|+C