Question

Given a function f:[a,b]→R and x∈R, prove that if the integral ∫ₐᵇ f(x)dx exists, then the integral ∫ₐᵇ f(x−c)axdx also exists, and these two integrals are equal.

Answer 1

To prove the equality of two integrals, we can use a change of variables. By substituting x = u + c and dx = du, we can show that the integral ∫ₐᵡₒ(x-c)axdx is equal to ∫ₐᵡₒ(u)axdu, and both integrals exist if the original integral ∫ₐᵡₒ(x)dx exists.

Step-by-step explanation:

To prove that if the integral ∫ₐᵡₒ(x)dx exists, then the integral ∫ₐᵡₒ(x-c)axdx also exists and these two integrals are equal, we can use a change of variables. Let u = x - c. Then x = u + c and dx = du. Substituting these expressions into the integral, we get:

∫ₐᵡₒ(x-c)axdx = ∫ₐᵡₒ(u)axdu

Since ∫ₐᵡₒ(x)dx exists, the integral ∫ₐᵡₒ(u)axdu also exists. And since x = a(u + c) remains in the interval [a, b] for the given u value, the limits of integration in both integrals remain the same. Therefore, these two integrals are equal.