required to prove that ∫f(x)dx[b, a] =∫f(x−c)dx [b+c, a+c]

jodecy

1. The problem statement, all variables and given/known data
required to prove that
∫𝑓(𝑥)𝑑𝑥[𝑏, 𝑎] =∫𝑓(𝑥−𝑐) 𝑑𝑥 [𝑏+𝑐, 𝑎+𝑐]

where f is a real valued function integrable over the interval [a, b]

2. Relevant equations

∫𝑓(𝑥)𝑑𝑥 [𝑏, 𝑎]=𝐹(𝑏)−𝐹(𝑎)

3. The attempt at a solution


∫𝑓(𝑥)𝑑𝑥 [b, a]=𝐹(𝑏)−𝐹(𝑎)

∫𝑓(𝑥−𝑐)𝑑𝑥 [𝑏+𝑐, 𝑎+𝑐]=𝐹(𝑏+𝑐−𝑐)−𝐹(𝑎+𝑐−𝑐)=𝐹(𝑏)−𝐹(𝑎)
∴∫𝑓(𝑥)𝑑𝑥[𝑏, 𝑎] =∫𝑓(𝑥−𝑐) 𝑑𝑥 [𝑏+𝑐, 𝑎+𝑐]

right i placed the interval in the [] brackets


is this correct?

Dick

You just assumed that the antiderivative of f(x-c) is F(x-c). Why is that true?

jodecy

i believe it was given in a lecture i had so i assumed is that a wrong assumption?

Dick

It's not a wrong assumption. It just needs to be proved. If F'(x)=f(x), why is F'(x-c)=f(x-c)? It's easy, but you should say why. Use the chain rule. In other language, they may expect you to prove this using the substitution u=x-c. Why is dx=du?

Borek

You are probably not aware, but the way you posted makes your post unreadable to at least XP Windows users using Chrome, IE & Opera, attachment shows what they see. It looks little bit better under Vista, but is still barely readable.

I have corrected thread subject.

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