Q. 78 Solve each of the integrals in E... [FREE SOLUTION]

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Chapter 5: Q. 78 (page 418)

Solve each of the integrals in Exercises 21鈥�70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)
${鈭�}_{0}^{1} x \sqrt{1 鈭� x} d x$

Short Answer

Expert verified

The solution of the given integral is ${鈭�}_{0}^{1} x \sqrt{1 鈭� x} d x = \frac{2}{3}$ .

Step by step solution

Step 1. Given Information

Solving the given integrals.

${鈭�}_{0}^{1} x \sqrt{1 鈭� x} d x$

Step 2. Using the substitution method.

Let

$u = 1 鈭� x \frac{d u}{d x} = 鈭� 1 d u = 鈭� d x 鈭� d u = d x$

Step 3. We will now write the limits of integration in terms of the new variable u.

When $x = 0$ , we have

$u = 1 鈭� x u = 1 鈭� 0 u = 1$

When role="math" localid="1649087278747" $x = 1$ , we have

$u = 1 鈭� x u = 1 鈭� 1 u = 0$

Step 4. Using the information in equations, we can change variables completely:

${鈭�}_{0}^{1} x \sqrt{1 鈭� x} d x = 鈭� {鈭�}_{1}^{0} \sqrt{u} d u {鈭�}_{0}^{1} x \sqrt{1 鈭� x} d x = 鈭� {鈭�}_{1}^{0} u^{1 / 2} d u {鈭�}_{0}^{1} x \sqrt{1 鈭� x} d x = 鈭� {[\frac{u^{1 / 2 + 1}}{1 / 2 + 1}]}_{1}^{0} {鈭�}_{0}^{1} x \sqrt{1 鈭� x} d x = 鈭� {[\frac{u^{3 / 2}}{3 / 2}]}_{1}^{0} {鈭�}_{0}^{1} x \sqrt{1 鈭� x} d x = 鈭� \frac{2}{3} {[u^{3 / 2}]}_{1}^{0} {鈭�}_{0}^{1} x \sqrt{1 鈭� x} d x = 鈭� \frac{2}{3} [{(0)}^{3 / 2} - {(1)}^{3 / 2}] {鈭�}_{0}^{1} x \sqrt{1 鈭� x} d x = 鈭� \frac{2}{3} [0 - 1] {鈭�}_{0}^{1} x \sqrt{1 鈭� x} d x = 鈭� \frac{2}{3} (- 1) {鈭�}_{0}^{1} x \sqrt{1 鈭� x} d x = \frac{2}{3}$

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91影视

Solve each of the integrals in Exercises 21鈥�70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)
${鈭�}_{0}^{1} x \sqrt{1 鈭� x} d x$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Using the substitution method.

Step 3. We will now write the limits of integration in terms of the new variable u.

Step 4. Using the information in equations, we can change variables completely:

One App. One Place for Learning.

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91影视

Solve each of the integrals in Exercises 21鈥�70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)鈭�01x1鈭�xdx

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Using the substitution method.

Step 3. We will now write the limits of integration in terms of the new variable u.

Step 4. Using the information in equations, we can change variables completely:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Pure Maths

Probability and Statistics

Theoretical and Mathematical Physics

Geometry

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Solve each of the integrals in Exercises 21鈥�70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)
${鈭�}_{0}^{1} x \sqrt{1 鈭� x} d x$