Q. 9 Why does it make sense that 鈭�0... [FREE SOLUTION]

Chapter 5: Q. 9 (page 477)

Why does it make sense that ${鈭�}_{0}^{1} \frac{1}{x^{p}} d x$ diverges when $p 鈮� 1$ ? Consider how $\frac{1}{x^{p}}$ compares with $\frac{1}{x}$ in this case.

Short Answer

Expert verified

For $p 鈮� 1$ , $\frac{1}{x^{p}}$ is greater than $\frac{1}{x}$ in the interval $[0, 1]$ , whose improper integral on $[0, 1]$ is known to diverge.

Step by step solution

Step 1. Given information.

We are given,

${鈭�}_{0}^{1} \frac{1}{x^{p}} d x$

Step 2. Comparing the integral

When the integral ${鈭�}_{0}^{1} \frac{1}{x^{p}} d x$ is solved in the interval $p 鈮� 1$ , it gives an result of $鈭�$ . Consider $\frac{1}{x}$ , since $p = 1$ .

Then,

{鈭�}_{0}^{1} \frac{1}{x} d x = [\ln x]_{0}^{1} = \ln 1 - \ln 0 = 鈭�

So, the integral is diverges.

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91影视

Why does it make sense that ${鈭�}_{0}^{1} \frac{1}{x^{p}} d x$ diverges when $p 鈮� 1$ ? Consider how $\frac{1}{x^{p}}$ compares with $\frac{1}{x}$ in this case.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Comparing the integral

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Why does it make sense that 鈭�011xpdxdiverges when p鈮�1? Consider how 1xpcompares with 1xin this case.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Comparing the integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Decision Maths

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Study anywhere. Anytime. Across all devices.

Why does it make sense that ${鈭�}_{0}^{1} \frac{1}{x^{p}} d x$ diverges when $p 鈮� 1$ ? Consider how $\frac{1}{x^{p}}$ compares with $\frac{1}{x}$ in this case.