Q.Evaluate $\displaystyle\int \frac{1}{4-x^{2}}\,dx$.

The denominator $4-x^{2}$ is in the $a^{2}-x^{2}$ form with $a=2$, so standard formula 5 applies directly.

Write the denominator as $2^{2}-x^{2}$, matching standard formula 5 with $a=2$:

$\int \frac{1}{4-x^{2}}\,dx = \frac{1}{2(2)}\ln\left|\frac{2+x}{2-x}\right|+C = \frac{1}{4}\ln\left|\frac{2+x}{2-x}\right|+C$ …

Applying formula 4 (for $x^{2}-a^{2}$) to a denominator that is actually written as $a^{2}-x^{2}$, which flips which term is subtracted inside the logarithm's fraction. Also, mis-identif …