   Chapter 4, Problem 9PS

Chapter
Section
Textbook Problem

Use the techniques of this chapter to verify this proposition.Proof Prove ∫ 0 x f ( t ) ( x − t ) d t = ∫ 0 x ( ∫ 0 t f ( v ) d v )   d t

To determine

To Prove: The specified integral 0xf(t)(xt)dt=0x(0tf(v)dv)dt.

Explanation

Given:

The specified integral is: 0xf(t)(xt)dt=0x(0tf(v)dv)dt

Formula used:

(uv)dx=vudu[ddxvvdv]dx

Proof:

Considering the integral,

I=0xf(t)(xt)dt

Since f(x) is anintegral function, hence, a function F(x) exists:

ddx[ F(x) ]=f(x)

That is,

F(x)=f(x)dx

Applying integration in parts:

f(t)(xt)dt=(xt)f(t)dt[ddt(xt)f(t)dt]dt=(xt)F(t)+F(t)dt

Hence,

0xf(t)(xt)dt=(xt)F(t)|0x+0xF(t)dt=(xx)F(x)(x0)F(0)+0xF(t)dt=xF(0)+0xF(t)dt

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