   Chapter 8, Problem 17PS

Chapter
Section
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:

uv=uv(v)u

Proof:

Take the left-hand side of the relation.

ab(xa)(xb)f(x)dx

Here u=(xa)(xb),v=f(x).

Use integration by parts.

ab(xa)(xb)f(x)dx={[(xa)(xb)f(x)dx]abab{(f(x)dx)ddx{(xa)(xb)}}dx}=[(xa)(xb)f(x)]abab{(xa)+(xb)}f(x)dx={[(ba)(bb)f(x)]ab[(aa)(a&#

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