Today we covered some tough examples of trigonometric substitution, for easier examples, read examples 1-7 in p.479 – p.482 of the textbook.
The examples I gave are unusual not in the techniques of trig sub itself, because the first step of identifying of the correct substitution is not really so hard. However, they are unusual second step, where we need to evaluate the weird trigonometric integrals that come out, and by now I have shown pretty much all special cases of $$\int \tan^m x \sec^n x dx$$. Note that the examples I gave doesn’t fall into the following cases
- m is not odd (otherwise, substitute \( u = \sec \theta\) will work)
- n is not even (otherwise, substitute \( u = \tan \theta\) will work)
Situations when m and n are super large numbers AND does not conform to the rules will MOST LIKELY NOT APPEAR again in MATH101, because it requires reduction formulas to be solved with a reasonable amount of work.
Similarly, for integrals of sines and cosines, when the powers of sine and cosine are odd, we can substitute cosine and sine respectively. But the case when both powers are even and large will most likely not appear again in this course. (for small even powers, though, we will use half angle formulas)
So before it confuses you finally, we isolate these really evil special cases:
- \( \int \sec x \ dx\)
- \( \int \sec^3 x dx\)
- \( \int \tan^2 x \sec x dx\)