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

The integral of 1/((x^2+3x+2)(x^2-1)) is 1/4 ln|x+1|+1/(2(x+1))-1/3 ln|x+2|+1/12 ln|x-1|+C

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

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

\frac{1}{4} ln ∣ x + 1 ∣ + \frac{1}{2 ( x + 1 )} - \frac{1}{3} ln ∣ x + 2 ∣ + \frac{1}{12} ln ∣ x - 1 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{1}{( x ^{2} + 3 x + 2 ) ( x ^{2} - 1 )} : \frac{1}{4 ( x + 1 )} - \frac{1}{2 ( x + 1 ) ^{2}} - \frac{1}{3 ( x + 2 )} + \frac{1}{12 ( x - 1 )}

= \int \frac{1}{4 ( x + 1 )} - \frac{1}{2 ( x + 1 ) ^{2}} - \frac{1}{3 ( x + 2 )} + \frac{1}{12 ( x - 1 )} 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}{4 ( x + 1 )} d x - \int \frac{1}{2 ( x + 1 ) ^{2}} d x - \int \frac{1}{3 ( x + 2 )} d x + \int \frac{1}{12 ( x - 1 )} d x

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

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

\int \frac{1}{3 ( x + 2 )} d x = \frac{1}{3} ln ∣ x + 2 ∣

\int \frac{1}{12 ( x - 1 )} d x = \frac{1}{12} ln ∣ x - 1 ∣

= \frac{1}{4} ln ∣ x + 1 ∣ - (- \frac{1}{2 ( x + 1 )}) - \frac{1}{3} ln ∣ x + 2 ∣ + \frac{1}{12} ln ∣ x - 1 ∣

Simplify

= \frac{1}{4} ln ∣ x + 1 ∣ + \frac{1}{2 ( x + 1 )} - \frac{1}{3} ln ∣ x + 2 ∣ + \frac{1}{12} ln ∣ x - 1 ∣

Add a constant to the solution

= \frac{1}{4} ln ∣ x + 1 ∣ + \frac{1}{2 ( x + 1 )} - \frac{1}{3} ln ∣ x + 2 ∣ + \frac{1}{12} ln ∣ x - 1 ∣ + C

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What is the integral of 1/((x^2+3x+2)(x^2-1)) ?
The integral of 1/((x^2+3x+2)(x^2-1)) is 1/4 ln|x+1|+1/(2(x+1))-1/3 ln|x+2|+1/12 ln|x-1|+C