   Chapter 8.8, Problem 15E

Chapter
Section
Textbook Problem

Evaluating an Improper Integral In Exercises 13-16, explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converges. ∫ 0 2 1 ( x − 1 ) 2 d x To determine

If the integral 021(x1)2dx is improper, will the integral converge and determine the value of the integral.

Explanation

Given:

The provided diagram is:

Explanation:

To check if the function is improper and determine the point of infinite discontinuity.

For this consider 1(x1)2= and find x:

1(x1)2=(x1)2=0x1=0x=1.

The point of discontinuity lies between upper limit and lower limit.

∴, it is concluded that given function is not continuous on interval [0,2].

Hence, given integral is improper.

Now since integral is discontinuous at x=1.

∴,

021(x1)2dx=011(x1)2dx+121(x1)2dx=limb1011(x1)2dx+limc1+121(x1)2dx

Integrate 1(x1)2:

limb1011(x1)2dx+limc1+121

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