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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.

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Introduction.

Given Information.

Explanation.

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Most popular questions from this chapter

Solve given definite integral.

$\int_{1}^{2} \frac{3}{{(9 - 2 x^{2})}^{3 / 2}} d x$

Solve $\int \frac{x}{4 + x^{2}} d x$ the following two ways:

(a) with the substitution $u = 4 + x^{2};$

(b) with the trigonometric substitution x = 2 tan u.

Why don’t we need to have a square root involved in order to apply trigonometric substitution with the tangent? In other words, why can we use the substitution $x = a \tan u$ when we see $x^{2} + a^{2}$ , even though we can’t use the substitution $x = a \sin u$ unless the integrand involves the square root of $a^{2} - x^{2}$ ? (Hint: Think about domains.)

Why is it okay to use a triangle without thinking about the unit circle when simplifying expressions that result from a trigonometric substitution with $x = a \sin u$ or $x = a \tan u$ ? Why do we need to think about the unit circle after trigonometric substitution with $x = a s e c u$ ?

Complete the square for each quadratic in Exercises 28–33. Then describe the trigonometric substitution that would be appropriate if you were solving an integral that involved that quadratic.

$x - 3 x^{2}$

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Draw pictures to illustrate why the comparison test for improper integrals makes intuitive sense for both convergence comparisons and for divergence comparisons.

Short Answer

Step by step solution

Introduction.

Given Information.

Explanation.

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

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