   Chapter 4, Problem 10PS

Chapter
Section
Textbook Problem

Use the techniques of this chapter to verify this proposition.Proof Prove ∫ a b f ( x ) f ' ( x ) d x = 1 2 ( [ f ( b ) ] 2 − [ f ( a ) ] 2 )

To determine

To Prove: The specified integral: abf(x)f(x)dx=12([ f(b) ]2[ f(a) ]2).

Explanation

Given:

The integral specified is: abf(x)f(x)dx=12([ f(b) ]2[ f(a) ]2)

Formula used:

xdx=x22+c

Proof:

Considering the integral,

I=abf(x)f(x)dx

Considering the transformations,

z=f(x)

Now, performing differentiation with regard to x as,

dz=f(x)dx

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