What is the integral of ((2x^2+1))/(x^3-3x+2) ?

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

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integral of ((2x^2+1))/(x^3-3x+2)

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

ln ∣ x - 1 ∣ - \frac{1}{x - 1} + ln ∣ x + 2 ∣ + C

Solution steps

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

Take the partial fraction of

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

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

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

\int \frac{1}{( x - 1 ) ^{2}} d x = - \frac{1}{x - 1}

\int \frac{1}{x + 2} d x = ln ∣ x + 2 ∣

= ln ∣ x - 1 ∣ - \frac{1}{x - 1} + ln ∣ x + 2 ∣

Add a constant to the solution

= ln ∣ x - 1 ∣ - \frac{1}{x - 1} + ln ∣ x + 2 ∣ + C

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