Q.Evaluate $\displaystyle\int \frac{1}{x^{2}-9}\,dx$.

Matching the denominator $x^{2}-9$ to the form $x^{2}-a^{2}$ gives $a=3$, so standard formula 4 applies directly.

Write the denominator as $x^{2}-3^{2}$, matching standard formula 4 with $a=3$:

$\int \frac{1}{x^{2}-9}\,dx = \frac{1}{2(3)}\ln\left|\frac{x-3}{x+3}\right|+C = \frac{1}{6}\ln\left|\frac{x-3}{x+3}\right|+C$ …

Using $2a=9$ instead of first identifying $a^{2}=9$ so $a=3$, which halves the coefficient incorrectly. Also confusing this with formula 5 (for $a^{2 …