Fundamental Theorem of Calculus (Part 2) Chapter 5.5 February 8, 2007 Fundamental Theorem of Calculus (Part 2) If f is continuous on [a, b], then : Where F is any antiderivative of f. ( )

Substitution Rule for Indefinite Integrals If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then Substitution Rule for Indefinite Integrals If g’(x) is continuous on [a,b] and f is continuous on the range of u = g(x), then

Using the Chain Rule, we know that: Evaluate:

Indefinite Integrals by Substitution Choose u. Try to choose u to be an inside function. (Think chain rule.) Try to choose u so that du is in the problem, except for a constant multiple. Calculate du. 2) Substitute u. Arrange to have du in your integral also. (All xs and dxs must be replaced!) 3) Solve the new integral. 4) Substitute back in to get x again.

Example A linear substitution:

Practice Let u =  du = 

Practice Let u =  du = 

Practice Let u = cos(x) du = –sin(x) dx An alternate possibility:

Practice Note:

Practice Let u =  du = 

Practice Let u =  du = 

Doesn’t Fit All We can’t use u–substitution to solve everything. For example: Let u = x2 du = 2x dx We need 2x this time, not just 2. We CANNOT multiply by a variable to adjust our integral. We cannot complete this problem

Doesn’t Fit All For the same reason, we can’t do the following by u–substitution: But we already knew how to do this!

Morals No one technique works for everything. Don’t forget things we already know! There are lots of integrals we will never learn how to solve… That’s when Simpson’s Rule, the Trapezoidal Rule, Midpoint,….. need to be used to estimate…...

Use u-substitution to find: Similarly for:

Definite Integral Examples:

Definite Integrals Evaluate:

Definite Integrals: Why change bounds? Can simplify calculations. (no need to substitute back to the original variable) Try:

Examples to try:

Summary Make a u-substitution: find u and du, then transform to an integral we can do. Be sure to transform back to the original variable! Rearrange to get du Substitution will still not solve every integral. (Nor is it always needed!)