Integral By Partial Fractions

Question

I. The denominator in the integrand $$\frac{1}{x^2-1}$$ factors as $$x^2-1 = (x-1)(x+1)$$.

II. Its partial fraction decomposition is given by $$\frac{1}{x^2-1} = \frac{1/2}{x-1} - \frac{1/2}{x+1}$$.

III. Integrating leads to the result $$\frac{1}{2}\ln\left|\frac{x-1}{x+1}\right| + C$$.

Which of the above statements is/are correct?
 
| Step                | Details                                               |
|---------------------|-------------------------------------------------------|
| Factor denominator  | $$x^2 - 1 = (x-1)(x+1)$$                              |
| Partial fractions    | $$\frac{1}{x^2-1} = \frac{1/2}{x-1} - \frac{1/2}{x+1}$$  |
| Integrate           | Leads to $$\frac{1}{2}\ln\left|\frac{x-1}{x+1}\right| + C$$ |

Accepted Answer

I, II, and III

Answer

II and III only

Answer

I and III only

Step	Details
Factor denominator	$$x^2 - 1 = (x-1)(x+1)$$
Partial fractions	$$\frac{1}{x^2-1} = \frac{1/2}{x-1} - \frac{1/2}{x+1}$$
Integrate	Leads to $$\frac{1}{2}\ln\left\|\frac{x-1}{x+1}\right\| + C$$

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