Trig Substitution

In trig substitution, we substitute $x$ and $\mathrm{d}x$ with trig functions of

Trig substitution can be appropriate when you have (in the integrand) one of these:

$\sqrt{a^2-x^2}$ - use $x=a\sin (\theta )$

$\sqrt{a^2+x^2}$ - use $x=a\tan (\theta )$

$\sqrt{x^2-a^2}$ - use $x=a\sec (\theta )$

Example:

$\int \frac{1}{(16-x^2{)}^{\frac{3}{2}}}$ $\int \frac{1}{(\sqrt{16-x^2}{)}^3}$