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

Problem

(∫_^)(1/(x2−1)*d(x))

Solution

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

1/(x2−1)=1/((x−1)*(x+1))

2. Decompose the integrand into partial fractions by setting up the equation for constants A and B

1/((x−1)*(x+1))=A/(x−1)+B/(x+1)

3. Solve for the constants by multiplying through by the common denominator (x−1)*(x+1)

1=A*(x+1)+B*(x−1)

4. Substitute x=1 to find A and x=−1 to find B

1=A(2)⇒A=1/2

1=B*(−2)⇒B=−1/2

5. Rewrite the integral using the partial fraction decomposition.

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

6. Integrate each term using the natural logarithm rule.

1/2*ln(x−1)−1/2*ln(x+1)+C

7. Simplify the expression using the properties of logarithms.

1/2*ln((x−1)/(x+1))+C

Final Answer

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

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