Integration by Substitution M.L.King Jr. Birthplace, Atlanta, GA Greg Kelly Hanford High School Richland, Washington Photo by Vickie Kelly, 2002

4.5 Day 1 Indefinite Integrals using Substitution Warm-Up Find the indefinite integral:

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!

The chain rule allows us to differentiate a wide variety of functions, but we are able to find antiderivatives for only a limited range of functions. We can sometimes use substitution to rewrite functions in a form that we can integrate.

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 2x in the original integrand. Many integrals can not be done by substitution.

Example 3: Solve for dx.

Example 4:

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

Example 6: You Try

Example 7: You Try

Example 8: You Try

Example 9: You Try (or WU Day 2)

AB Homework for Section 4.5 Day Pg. 304 #7-37 odd

BC Homework for Section 4.5 Day Pg. 304 #7-37 odd, odd

4.5 Day 2 Indefinite Integrals using Substitution

Example 1: You Try

Example 2: You Try

Example 3: The technique is a little different for definite integrals. We can find new limits, and then we don’t have to substitute back. new limit We could have substituted back and used the original limits.

Example 3 (con’t.): Wrong! The limits don’t match! Using the original limits: Leave the limits out until you substitute back. This is usually more work than finding new limits

Example 4: Don’t forget to use the new limits.

Example 5: You Try

Integration of Even and Odd Functions A function is odd if Polynomials are odd if each term has an odd exponent. A function is even if Polynomials are even if each term has an even exponent or is a constant.

Example Evaluate Solution: Letting f(x) = sin 3 x cos x + sin x cos x produces f(–x) = sin 3 (–x) cos (–x) + sin (–x) cos (–x) = –sin 3 x cos x – sin x cos x = –f(x).

Solution So, f is an odd function, and because f is symmetric about the origin over [–π/2, π/2], you can apply Theorem 4.16 to conclude that Figure 4.41

Example Solution: Since the integrand is an even function and it is being integrated with symmetry about the origin, you can rewrite it as: Evaluate the integral. Evaluate

Splitting an integral to use even/odd properties Solution: Split the integral to create an odd function and an even function, each with symmetry about the origin. Evaluate the integrals, add the results, and check the answer on your calculator. Evaluate

AB Homework for Section 4.5 Day p ,odd 71,73, odd.

BC Homework for Section 4.5 Day p odd, odd, 87, 89, odd

Homework for Section 4.5 Day 3 MMM pgs. 158, 159 Quiz tomorrow on 4.5 and Barron’s Ch.2,3

4.5 Day 2 Definite Integrals using Substitution Change of Variables Calculus HWQ 12/3 Find an equation for the function f that has the derivative and whose graph passes through the point (0, 3).

Calculus HWQ 11/19 No calculator please Evaluate the integral:

Calculus Warm-up 12/6 Find the indefinite integral (#69):

AB Homework for Section 4.5 Day 4 Pg odd