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## Homework Statement

Suppose that f is a continuous function.

a) If g is a differentiable function and if

F(x) = [tex]\int_{0}^{x}[/tex]g(x)f(t)dt

find F ' (x).

b) Show that

[tex]\int_{0}^{x}[/tex](x - t)f(t)dt = [tex]\int_{0}^{x}[/tex]

**[**[tex]\int_{0}^{u}[/tex]f(t)dt

**]**du

Suggestion: Use a) and the racetrack theorem.

## Homework Equations

The fundamental theorems of calculus.

The racetrack theorem states that if

f(a) = g(a) and if f ' (x) = g ' (x) for all x, then f(x) = g(x) for all x.

## The Attempt at a Solution

a)

I took d/dx of both sides and then ended up with

F ' (x) = g(x)f(x) by the first fundamental theorem.

b)

I have to prove that the left side and right side are equal at some point a, and that their derivatives are equal everywhere. I've never come across the integral of an integral before so I'm not quite sure what to do.