Chapter 8: Q. 62 (page 702)

Use the results from Exercises 51–60 and Theorem 7.38 to approximate the values of the definite integrals in Exercises 61–70 to within 0.001 of their values.
$\int_{0}^{1} x \cos (x^{3}) d x$

Short Answer

Expert verified

The approximate value is $0 \cdot 06752$ .

Step by step solution

Step 1. Given information .

Consider the given integral $\int_{}^{1} x \cos (x^{3}) d x$ .

Step 2. Using the result from Exercises 51–60 and Theorem 7.38 .

The result of $\cos x = \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{(2 k)} x^{2 k}$

Theorem 7.38 - Let L be the sum of an alternating series satisfying the hypotheses of the alternating series test. For any term Sn in the sequence of partial sums,. Furthermore, the sign of the difference L − Sn is the sign of the coefficient of the term .

Step 3. Find the value .

$\int_{0}^{1} x \cos (x^{3}) d x = \int_{0}^{1} x \cdot \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{(2 k)} {(x^{3})}^{2 k} d x = \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{(2 k)} \int_{0}^{1} x x^{6 k} d x = \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{(2 k)} \int_{0}^{1} x^{6 k + 1} d x = \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{(2 k)} {[\frac{x^{6 k + 2}}{6 k + 2}]}_{0}^{1} = \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{(2 k)} [\frac{1}{6 k + 2}]$

Substitute $k = 0, 1, 2, 3 . . . . . . . . \infty$

$\Rightarrow \frac{1}{8} - \frac{1}{16} + \frac{1}{192} - \frac{1}{5760} + . . . . . . . . . . . . \Rightarrow 0 \cdot 125 - 0 \cdot 0625 + 0 \cdot 00520 - 0 \cdot 0001736 \Rightarrow 0 \cdot 06752$

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Use the results from Exercises 51–60 and Theorem 7.38 to approximate the values of the definite integrals in Exercises 61–70 to within 0.001 of their values.
$\int_{0}^{1} x \cos (x^{3}) d x$

Short Answer

Step by step solution

Step 1. Given information .

Step 2. Using the result from Exercises 51–60 and Theorem 7.38 .

Step 3. Find the value .

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Use the results from Exercises 51–60 and Theorem 7.38 to approximate the values of the definite integrals in Exercises 61–70 to within 0.001 of their values.∫01xcosx3dx

Short Answer

Step by step solution

Step 1. Given information .

Step 2. Using the result from Exercises 51–60 and Theorem 7.38 .

Step 3. Find the value .

One App. One Place for Learning.

Most popular questions from this chapter

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Study anywhere. Anytime. Across all devices.

Use the results from Exercises 51–60 and Theorem 7.38 to approximate the values of the definite integrals in Exercises 61–70 to within 0.001 of their values.
$\int_{0}^{1} x \cos (x^{3}) d x$