Question

Let $ T>0 $ be a fixed number. If $ f: \mathbb{R} \to \mathbb{R} $ is a continuous function such that $ f(x + T) = f(x) $, then: $$ \text{If } I = \int_0^T f(x)\, dx,\quad \text{then } \int_0^{5T} f(2x)\, dx = ? $$

• A. \( 10I \)
• B. \( \frac{5}{2} I \)
• C. \( 5I \)
• D. \( 2I \)

Hint:

Always consider substitution when evaluating \( f(ax) \) inside integrals, especially when \( f \) is periodic.

Answer 1

The Correct Option is C

We are given that \( f(x + T) = f(x) \Rightarrow \text{periodic with period } T \) We need to evaluate: \[ \int_0^{5T} f(2x) dx \Rightarrow \text{Use substitution: let } u = 2x \Rightarrow dx = \frac{du}{2} \Rightarrow x = 0 \Rightarrow u = 0,\quad x = 5T \Rightarrow u = 10T \] So: \[ \int_0^{5T} f(2x) dx = \int_0^{10T} f(u) \cdot \frac{1}{2} du = \frac{1}{2} \int_0^{10T} f(u) du \] But \( f \) is periodic with period \( T \Rightarrow \int_0^{10T} f(u)\, du = 10I \) So: \[ \frac{1}{2} \cdot 10I = 5I \Rightarrow \boxed{5I} \]