Question

Evaluate \[\int \frac{dx}{a^2-x^2}\].

Options

• \[\log\left|x+\sqrt{x^2-a^2}\right|+C\]

• \[\frac{1}{2a}\log\left|\frac{a+x}{a-x}\right|+C\]

• \[\frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right)+C\]

• \[\frac{1}{2a}\log\left|\frac{x-a}{x+a}\right|+C\]

Answer 1

For the standard form \[a^2-x^2\], the logarithmic ratio is \[\frac{a+x}{a-x}\]. The factor multiplying the logarithm is \[\frac{1}{2a}\].