Chapter 5: Q. 77 (page 418)

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)
$\int_{0}^{1} \frac{x}{1 + x^{4}} d x$

Short Answer

Expert verified

The solution of the given integral is $\int_{0}^{1} \frac{x}{1 + x^{4}} d x = \frac{π}{8}$ .

Step by step solution

Step 1. Given Information

Solving the given integrals.

$\int_{0}^{1} \frac{x}{1 + x^{4}} d x$

Step 2. Using the substitution method.

$\int_{0}^{1} \frac{x}{1 + x^{4}} d x = \int_{0}^{1} \frac{x}{1 + {(x^{2})}^{2}} d x$

Let

role="math" localid="1649086854342" $u = x^{2} \frac{d u}{d x} = 2 x d u = 2 x d x \frac{1}{2} d u = x d x$

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

When $x = 0$ , we have

$u = x^{2} u = 0^{2} u = 0$

When role="math" localid="1649086870398" $x = 1$ , we have

$u = x^{2} u = {(1)}^{2} u = 1$

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

$\int_{0}^{1} \frac{x}{1 + x^{4}} d x = \frac{1}{2} \int_{0}^{1} \frac{1}{1 + u^{2}} d u \int_{0}^{1} \frac{x}{1 + x^{4}} d x = \frac{1}{2} {[\tan^{- 1} x]}_{0}^{1} \int_{0}^{1} \frac{x}{1 + x^{4}} d x = \frac{1}{2} [\tan^{- 1} 1 - \tan^{- 1} 0] \int_{0}^{1} \frac{x}{1 + x^{4}} d x = \frac{1}{2} [\frac{π}{4} - 0^{o}] \int_{0}^{1} \frac{x}{1 + x^{4}} d x = \frac{1}{2} \cdot \frac{π}{4} \int_{0}^{1} \frac{x}{1 + x^{4}} d x = \frac{π}{8}$

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Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)
$\int_{0}^{1} \frac{x}{1 + x^{4}} d x$

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:

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Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)∫01x1+x4dx

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

Theoretical and Mathematical Physics

Statistics

Decision Maths

Mechanics Maths

Applied Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)
$\int_{0}^{1} \frac{x}{1 + x^{4}} d x$