4.5 Integration by substitution

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