Q.Evaluate $\displaystyle\int \frac{dx}{x^{2}+9}$.

The denominator $x^{2}+9$ is in the form $x^{2}+a^{2}$ with $a=3$, so standard form 3 (the inverse-tangent form) applies. …

Write the denominator as $x^{2}+3^{2}$, matching standard form 3 with $a=3$:

Check by differentiation: using $\dfrac{d}{dx}\tan^{-1}u=\dfrac{1}{1+u^{2}}\cdot\dfrac{du}{dx}$ with $u=\dfrac{x}{3}$, …

Mixing up the $+$ and $-$ forms: a $x^{2}+a^{2}$ denominator gives an inverse-tangent, while $x^{2}-a^{2}$ gives a logarithm. Also writing the coefficient as $\frac{1}{2a}$ (which be …