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Chapter 5: Q. 39 (page 478)

Use limits of definite integrals to calculate each of the improper integrals in Exercises.
${鈭�}_{0}^{鈭�} \frac{1}{\sqrt{x} e^{\sqrt{x}}} d x$

Short Answer

Expert verified

The improper integral on $[0, 鈭�)$ converges.

${鈭�}_{0}^{鈭�} \frac{1}{\sqrt{x} e^{\sqrt{x}}} d x = 2 .$

Step by step solution

01

Step 1. Given information.

The given integral is the following.

${鈭�}_{0}^{鈭�} \frac{1}{\sqrt{x} e^{\sqrt{x}}} d x$

02

Step 2. Value of integral.

Solve the integral by using the definition of improper integral.

${鈭�}_{0}^{鈭�} \frac{1}{\sqrt{x} e^{\sqrt{x}}} d x = {鈭�}_{0}^{鈭�} x^{\frac{- 1}{2}} e^{- \sqrt{x}} d x = \lim_{b 鈫� 鈭�} {鈭�}_{0}^{b} x^{\frac{- 1}{2}} e^{- \sqrt{x}} d x = \lim_{b 鈫� 鈭�} {[2 e^{- \sqrt{x}} x^{\frac{1}{2}} - (- 2 (- e^{- \sqrt{x}} \sqrt{x} - e^{- \sqrt{x}}))]}_{0}^{b} = \lim_{b 鈫� 鈭�} {[2 e^{- \sqrt{x}} \sqrt{x} - 2 e^{- \sqrt{x}} \sqrt{x} - 2 e^{- \sqrt{x}}]}_{0}^{b} = \lim_{b 鈫� 鈭�} {[- 2 e^{- \sqrt{x}}]}_{0}^{b} = \lim_{b 鈫� 鈭�} {[- 2 e^{- \sqrt{b}} - (- 2)]}_{0}^{b} = 0 + 2 = 2$

The value of the integral is ${鈭�}_{0}^{鈭�} \frac{1}{\sqrt{x} e^{\sqrt{x}}} d x = 2 .$

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Most popular questions from this chapter

Solve the integral: $鈭� x \ln \sqrt{x} d x$

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

$2 x^{2} 鈭� 4 x + 1$

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 $x = \tan 鈦� u$ .

(b) An integral with which we could reasonably apply trigonometric substitution with $x = 4 \sec 鈦� u$ .

(c) An integral with which we could reasonably apply trigonometric substitution with $x 鈭� 2 = 3 \sin 鈦� u$ .

Explain why, if $x = a s e c u$ , then $\sqrt{a^{2} s e c^{2} u 鈭� a^{2}}$ is $鈭� a \tan u$ if $x < 鈭� a$ and is $a \tan u$ if $x > a$ . Your explanation should include a discussion of domains and absolute values.

Consider the integral $鈭� x {(x^{2} 鈭� 1)}^{2} d x$ .

(a) Solve this integral by using u-substitution.

(b) Solve the integral another way, using algebra to multiply out the integrand first.

(c) How must your two answers be related? Use algebra to prove this relationship.

See all solutions

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