Chapter 8, Problem 17PS

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

# Proof Suppose that f ( a ) = f ( b ) = 0 and the second derivatives of f exist on the closed interval [ a ,   b ] . Prove that ∫ a b ( x − a ) ( x − b ) f ″ ( x ) d x = 2 ∫ a b f ( x ) d x .

To determine

To prove: ab(xa)(xb)f(x)dx=2abf(x)dx.

Explanation

Given:

The functions f(a)=f(b)=0 and the second derivatives of f is continuous on the interval [a,b].

Formula used:

Integration by parts:

âˆ«uâ‹…v=uâˆ«vâˆ’âˆ«(âˆ«v)uâ€²

Proof:

Take the left-hand side of the relation.

âˆ«ab(xâˆ’a)(xâˆ’b)fâ€²â€²(x)dx

Here u=(xâˆ’a)(xâˆ’b),v=fâ€²â€²(x).

Use integration by parts.

âˆ«ab(xâˆ’a)(xâˆ’b)fâ€²â€²(x)dx={[(xâˆ’a)(xâˆ’b)âˆ«fâ€²â€²(x)dx]abâˆ’âˆ«ab{(âˆ«fâ€²â€²(x)dx)â‹…ddx{(xâˆ’a)(xâˆ’b)}}dx}=[(xâˆ’a)(xâˆ’b)fâ€²(x)]abâˆ’âˆ«ab{(xâˆ’a)+(xâˆ’b)}fâ€²(x)dx={[(bâˆ’a)(bâˆ’b)fâ€²(x)]abâˆ’[(aâˆ’a)(a&#

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