1. 4.5 Integration by substitution
Use pattern recognition to find an indefinite integral.
Use a change of variables to find an indefinite integral.
Use the general power rule for integration to find an indefinite integral.
Use a change of variables to evaluate a definite integral.

2. Two techniques for integrating composite functions, pattern recognition & change of variables. I am going to teach you the change of variables.
This concept is comparable to the role of the chain rule in differentiation.
y = f(g(x))
y’ = f’(g(x)) • g’(x)
Antidifferentiation of a composite function
∫f(g(x))g’(x) dx = F(g(x)) + c
If u = g(x), then du = g’(x) dx and
∫f(u)du = F(u) + c = F(g(x)) + c

3. The difficult part of using the theorem is deciding which function to make g’(x) and that comes from knowing your derivatives and experience.
∫2x(x² + 1)⁴dx 2) ∫3x²√x³ +1 dx
3) ∫sec²x(tanx + 3)dx 4)∫x(x² + 1)²dx

5. Change of variables- guidelines (p. 299)

6. General Power Rule for Integration

7. Change of variables for definite integrals
2 methods