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Chapter 5: Q. 75 (page 479)

Prove each statement in Exercises 74鈥�77, using limits of definite integrals for general values of p.

If p > 1, then鈭�1鈭�1xpdxconverges to1p-1.

Short Answer

Expert verified

The given statement is proved.

Step by step solution

01

Step 1. Given Information.

The given integral is鈭�1鈭�1xpdx.

02

Step 2. Prove. 

To prove if p > 1, then 鈭�1鈭�1xpdxconverges to 1p-1, let aand bbe any real numbers and f(x) be a function, and if fcontinuous on role="math" localid="1649067096801" [a,鈭�),then 鈭�a鈭�f(x)dx=limB鈫�鈭�鈭�aBf(x)dx.

So,

鈭�1鈭�1xpdx=limB鈫�鈭�鈭�1Bx-pdx=limB鈫�鈭�x-p+1-p+11B=limB鈫�鈭�B1-p1-p-11-p=0-11-p=-1-p-1=1p-1

Now, p > 1, so the given integral converges to 1p-1.

Hence, the integral is proved.

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