7.5: Strategies for Integration

Practice: 7.5.2: p. 510-511 #23 - 25 all, 30, 31.

Note the Table of Integration Formulas on page 506. The only ones we have not learned are the hyperbolic functions (note their similarity to the trig functions): $LaTeX: \int\sinh\left(x\right)dx\:=\:\cosh\left(x\right)\:\:and\:\:\int\cosh\left(x\right)dx\:=\:\sinh\left(x\right)$

This section summarizes and synthesizes all the techniques you have learned for integration.

Summarizing their recommendations:

Simplify the integrand if possible.
Look for an obvious substitution.
Classify the integrand according to its form.
1. Trig Functions
2. Rational Functions
3. Integration by Parts
4. Radicals
[My favorite. . . ] Try Again!!!

As they point out, there are really only two methods available to you: substitution and parts.

This is where you get to be creative. Try different approaches.

Multiply by a helpful form of 1.
If you wish there were something added on, try adding zero (by adding and subtracting the same thing).
Squint at it and see if it looks "familiar" (like you've solved something like it before) and see if an old technique will work in a new situation.
Try combining methods.

Notice, in one of my favorite phrases in the text, Stewart tells you "the functions that we have been dealing with in this book are called elementary functions." (I know, they don't feel very elementary do they? But by the definition of elementary, they are.)

He goes on to say that most elementary functions don't actually have elementary antiderivatives which is, in part, why we need to be able to express elementary functions as series -- the longer term goal of the BC curriculum.