Definite Integration Question - None

Question

Given $ g(x) = f(x+5) $, $ f(0) = 0 $, and $ g(x) $ is an even function. Find $ I = \int_0^x f(t)\,dt $ in terms of $g$.

Accepted Answer

Step 1: Use the even function property of g(x). Since g(x) = f(x+5) is even:

g(-x) = g(x) ⟹ f(-x+5) = f(x+5) ⟹ f(5-x) = f(5+x)

This shows f is symmetric about x = 5.

Step 2: Apply the symmetry condition. From f(5-x) = f(5+x), substitute u = 5+x:

f(5-(u-5)) = f(u) ⟹ f(10-u) = f(u)

Step 3: Use the boundary condition f(0) = 0 and the symmetry f(10-x) = f(x). This means f(10) = f(0) = 0.

Step 4: For the integral ∫₀ˣ f(t)dt, substitute t = 5+s in the integral of g:

∫₋₅^(x-5) g(s)ds = ∫₋₅^(x-5) f(s+5)ds = ∫₀ˣ f(t)dt

Using even property of g and limits adjustment:

I = ∫₀ˣ f(t)dt = ∫₀^(x-5) g(s)ds + boundary terms

Step 5: Express in standard form. The relationship is:

∫₀ˣ f(t)dt = ∫₀^(x-5) g(u)du (after careful limit transformation)

∴ Answer: A

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