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Chapter 5: Q.10 (page 477)
Draw pictures to illustrate why the comparison test for improper integrals makes intuitive sense for both convergence comparisons and for divergence comparisons.
1
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Explain why using trigonometric substitution with often involves a triangle with side lengths a and x and hypotenuse of length
Explain how to use long division to write the improper fraction as the sum of an integer and a proper fraction.
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?
Solve the integral:.
Explain how to know when to use the trigonometric substitutions , Describe the trigonometric identity and the triangle that will be needed in each case. What are the possible values for and in each case?
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