Chapter 8: Q. 64 (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.
$\int_{0 \cdot 1}^{0 \cdot 2} \tan^{- 1} (x^{2}) d x$

Short Answer

Expert verified

The approximate value is 0.238 .

Step by step solution

Step 1. Given information .

Consider the given integral $\int_{0 \cdot 1}^{0 \cdot 2} \tan^{- 1} (x^{2}) d x$ .

Step 2. Using the result of tan-1x from question 51 - 60 and theorem 7.38 .

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

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 .

$\int_{0 \cdot 1}^{0 \cdot 2} \tan^{- 1} (x^{2}) d x = \int_{0 \cdot 1}^{0 \cdot 2} \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{2 k + 1} x^{2 (2 k + 1)} d x = \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{2 k + 1} \int_{0 \cdot 1}^{0 \cdot 2} x^{4 k + 2} d x = \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{2 k + 1} {[\frac{x^{4 k + 3}}{4 k + 3}]}_{0 \cdot 1}^{0 \cdot 2} = \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{2 k + 1} [\frac{0 \cdot 2^{4 k + 3} - 0 \cdot 1^{4 k + 3}}{4 k + 3}]$

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

role="math" localid="1649681729809" $\Rightarrow \frac{7}{30} + \frac{0 \cdot 0142}{3} \Rightarrow 0 \cdot 2333 + 0 \cdot 0047 \Rightarrow 0 \cdot 238$

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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.
$\int_{0 \cdot 1}^{0 \cdot 2} \tan^{- 1} (x^{2}) d x$

Short Answer

Step by step solution

Step 1. Given information .

Step 2. Using the result of tan-1x from question 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.∫0·10·2tan-1x2dx

Short Answer

Step by step solution

Step 1. Given information .

Step 2. Using the result of tan-1x from question 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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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.
$\int_{0 \cdot 1}^{0 \cdot 2} \tan^{- 1} (x^{2}) d x$