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 387]. 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 389, Example 6 page 390]. Exercises 21, 25, 53.
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 394; 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, Exercises 21, 23].
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 357]. 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. [Figures 2, 3 page 413] Understand how to compute the Midpoint approximation M and how to apply Simpson's Rule [Box, page 418, Example 4] Exercise 25.
5.10 Improper integrals. A: Infinite interval of integration.
Understand how to calculate an integral from a to
infinity as the limit, as T goes to infinity, of the
integral from a to T if that limit exists.
[Fundamental examples 1/x2 on page 423 and
1/x in Example 1]. Understand why 1/xp
gives a convergent integral from 1 to infinity if p > 1
and a divergent integral (no limit) otherwise [Example 4].
B: Function goes to infinity at a finite value a. Understand how to calculate an integral from a to b as the limit, as t goes to a, of the integral from t to b if that limit exists. Fundamental examples 1/x1/2 [See Example 5] and 1/x. Understand why 1/xp gives a convergent integral from 0 to 1 if p < 1 and a divergent integral (no limit) otherwise. Exercises 23, 25. Be able to apply the Comparison Test [Example 9] Exercises 17, 19.