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Chapter 5: Q. 40 (page 417)
Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)
The solution of the given integral is .
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Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.
(a) An integral with which we could reasonably apply trigonometric substitution with .
(b) An integral with which we could reasonably apply trigonometric substitution with .
(c) An integral with which we could reasonably apply trigonometric substitution with .
Show by differentiating (and then using algebra) that and are both antiderivatives of . How can these two very different-looking functions be an antiderivative of the same function?
Find three integrals in Exercises 21–70 in which the denominator of the integrand is a good choice for a substitution u(x).
Suppose v(x) is a function of x. Explain why the integral
of dv is equal to v (up to a constant).
Find three integrals in Exercises 21–70 that we can anti-differentiate immediately after algebraic simplification.
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