Some tricky integrals can be simplified significantly by the use of trigonometric substitutions. Any time you see one of the expressions in the left column, you make the substitution and look for the related Identity # Trig Sub Table | Expression | Substitution | Related Identity | | ----------- | ------------- | ------------------------- | | $a^2-x^2$ | $x=a \sin u$ | $1-\sin^2 u=\cos^2 u$ | | $x^2-a^2$ | $x=a\sec u$ | $\sec^2 u-1=\tan^2 u$ | | $a^2 +x$ | $x=a\tan u$ | $1+\tan^2 u=\sec^2 u$ | | $x^2-a^2$ | $x=a \cosh u$ | $\cosh^2 u-1 = \sinh^2 u$ | | $a^2 + x^2$ | $x=a \sinh u$ | $1+\sinh^2 u = \cosh^2 u$ | # Process To solve these problems: 1. Identify some expression in $x$ which matches the "expression" column of the table 2. set $x$ to your substitution 3. differentiate both sides to find $\mathrm{d}x$ 4. Substitute for $x$ and $\mathrm{d}x$ in the expression 5. Simplify. You will likely find the problem has been engineered such that the "Related Identity" is helpful at some point. Notice sometimes you will have a choice of a hyperbolic or a regular trig substitution. In those cases you can technically do either and the answer will be the same, but generally if there isn't one which is obviously the correct choice then it probably makes absolutely no difference which you pick. If you check with a CAS, you will find that (although superficially appearing different) the final expressions are identically equal (assuming you've done everything correctly of course). The way you would know which was "obviously the correct choice" was if one or other of the "related identities" shows up somehow. # Further Exploration Blackpenredpen has an excellent trig substitution playlist