5.5 Substitution rule. Know how to recognize the "outer" function f and the "inner" function g in the integrand f(g(x)) g'(x) dx, and how to then simplify the integrand by rewriting it as f(u)du. [Example 1 page 390]. Remember, in an indefinite integral, to return to the original variable (in this case, x) for your answer. Know how to transform the limits of integration in a definite integral [Equation 5 page 392, Example 6 page 393]. Exercises 7, 24, 43.
5.6 Integration by parts. Know how to choose u and dv so that v du will be an easier integrand than u dv. [Note at bottom of page 397; Exercises 3,5]. Know when to do two consecutive integrations by parts [Example 3, Exercise 7]. Know the method for treating integrands like ex sin x dx [Example 4, Exercise 13]. Know the "exotic" integrations by parts: ln x dx, arctan x dx [Examples 2, 5, Exercise 6].
5.9 Approximate integration. Know how to carry out a left endpoint approximation L ("left-hand sum") for a definite integral, given a number n of (equal) subintervals. Know also how to compute the "right-hand sum" R [Example 2(a) page 360]. Know that if f is increasing on an interval [a,b] then L underestimates, and R overestimates, the integral of f from a to b. Understand how to compute the Trapezoidal approximation T = (L + R)/2 and that T overestimates the integral if f is concave up, and underestimates if f is concave down. [Figure 5 page 419]. Understand how to compute the Midpoint approximation M and how to apply Simpson's Rule S = (T + 2M)/3. Exercise 23.