CHAPTER 4 SECTION 4 5 INTEGRATION BY SUBSTITUTION

CHAPTER 4 SECTION 4. 5 INTEGRATION BY SUBSTITUTION

Theorem 4. 12 Antidifferentiation of a Composite Function

Substitution with Indefinite Integration • This is the “backwards” version of the chain rule • Recall … • Then …

Substitution with Indefinite Integration • In general we look at the f(x) and “split” it – into a g(u) and a du/dx • So that …

Substitution with Indefinite Integration • Note the parts of the integral from our example

Substitution with Indefinite Integration • Let u = So, du = (2 x -4)dx

Guidelines for Making a Change of Variables

Theorem 4. 13 The General Power Rule for Integration

Example 1: The variable of integration must match the variable in the expression. Don’t forget to substitute the value for u back into the problem!

Example 2: One of the clues that we look for is if we can find a function and its derivative in the integrand. The derivative of is Note that this only worked because of the 2 x in the original. Many integrals can not be done by substitution. .

Example 3: Solve for dx.

Example 4:

Example 5: We solve for because we can find it in the integrand.

Example 6:

Can You Tell? • Which one needs substitution for integration?

Integration by Substitution

Integration by Substitution

Solve the differential equation

Solve the differential equation

Theorem 4. 14 Change of Variables for Definite Integrals

or you could convert the bound to u’s.

Example 7: new limit The technique is a little different for definite integrals. We can find new limits, and then we don’t have to substitute back.

Example 9: Don’t forget to use the new limits.

Theorem 4. 15 Integration of Even and Odd Functions

Even/Odd Functions If f(x) is an even function, then If f(x) is an odd function, then

Even/Odd Functions If f(x) is an even function, then If f(x) is an odd function, then