Techniques of integration (9/5/08) Finding derivatives involves facts and rules; it is a completely mechanical process. Finding antiderivatives is not completely mechanical. It involves some facts, a couple of rules, and then various techniques which may or may not work out. There are many functions (e.g., f(x) = e x^2 ) which have no known antiderivative formula.

There Are a Couple of “Rules” Sum and Difference Rule: Antiderivatives can be found working term by term (just like derivatives). Constant Multiplier Rule: Constant multipliers just get carried along as you get antiderivatives (just like derivatives). HOWEVER, there is no Product Rule, Quotient Rule, or Chain Rule for Antiderivatives!

Reversing the Chain Rule: “substitution” or “guess and check” Any ideas about  x 2 (x 3 + 4) 5 dx ?? How about  x e x^2 dx ? Try  ln(x) / x dx But we’ve been lucky! Try  sin(x 2 ) dx

The Substitution Technique It’s called a “technique”, not a “rule”, because it may or may not work. If there is a chunk, try calling the chunk u. Compute du = (du/dx) dx Replace all parts of the original expression with things involving u (i.e., eliminate x). If you were lucky/clever, the new expression can be anti-differentiated easily.

Assignment for Monday We will have a regular class on Monday. Read Section 5.5 of the text and go over today’s class notes. In Section 5.5, do Exercises 1-31 odd.