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Chapter 5: Q. 8 (page 477)

Why does it make sense that 鈭�1鈭�1xpdx diverges when 0<p鈮�1 ? Consider how 1xp compares with 1x in this case.

Short Answer

Expert verified

The integral gives the indefinite result forp=1.

Step by step solution

01

Step 1. Given information.   

We are given,

鈭�1鈭�1xpdx

The integral diverges when 0<p鈮�1.

02

Step 2. Comparing the integral

When the integral 鈭�1鈭�1xpdxis solved in the interval 0<p鈮�1, it gives an result of 鈭�.

As when p=1is considered in the interval.

Then the resulting integral is lnxwhere the interval [鈭�,1]is applied which gives the indefinite result.

That's why the result is diverges.

In this case, 1xpis equal to 1xwhen p=1.

So,

鈭�1鈭�1xdx=[lnx]1x=ln鈭�-ln1=鈭�

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