   Chapter 4.5, Problem 102E

Chapter
Section
Textbook Problem

Rewriting Integrals Show that if f is continuous on the entire real number line, then ∫ 0 b f ( x + h )   d x = ∫ a + h b + h f ( x ) d x

To determine

To prove: The equation: abf(x+h)dx=a+hb+hf(x)dx

Explanation

Given: Function, f(x) is continuous on entire real number line, and the equation: abf(x+h)dx=a+hb+hf(x)dx.

Formula used:

On the closed interval [a,b], if the function, u=g(x) has a continuous derivative and f is continuous on the range of g, then

abf(g(x))g(x)dx=g(a)g(b)f(u)du

Proof:

Let, u=x+h, and differentiate it with respect to x,

u=x+hdu=dx

For x=a, the value of u=a+h

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