Evaluate the Integral integral of 2/(x^2-1) with respect to x

Problem

(∫_^)(2/(x2−1)*d(x))

Solution

1. Factor the denominator using the difference of squares formula.

(∫_^)(2/((x−1)*(x+1))*d(x))

2. Decompose the integrand into partial fractions by setting the expression equal to a sum of simpler fractions.

2/((x−1)*(x+1))=A/(x−1)+B/(x+1)

3. Solve for the constants A and B by multiplying both sides by the common denominator.

2=A*(x+1)+B*(x−1)

4. Substitute x=1 to find A

2=A*(1+1)⇒A=1

5. Substitute x=−1 to find B

2=B*(−1−1)⇒B=−1

6. Rewrite the integral using the partial fraction decomposition.

(∫_^)((1/(x−1)−1/(x+1))*d(x))

7. Integrate each term separately using the natural logarithm rule.

ln(x−1)−ln(x+1)+C

8. Simplify the expression using the properties of logarithms.

ln((x−1)/(x+1))+C

Final Answer

(∫_^)(2/(x2−1)*d(x))=ln((x−1)/(x+1))+C

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