Let `T >0` be a fixed real number. Suppose `f` is continuous function such that for all `x in R ,f(x+T)=f(x)dot` If `I=int_0^Tf(x)dx ,` then the value of `int_3^(3+3T)f(2x)dx` is (a)`3/2I` (b) `2I` (c) `3I` (d) `6I`

Options

A. `3/2I`
B. `2I`
C. `3I`
D. `6I`

Explanation

To solve the problem, we need to evaluate the integral \( \int_{3}^{3+3T} f(2x) \, dx \) given that \( f(x+T) = f(x) \) for all \( x \in \mathbb{R} \) and \( I = \int_{0}^{T} f(x) \, dx \).

### Step-by-Step Solution:

1. **Substitution**: Let \( y = 2x \). Then, differentiating both sides gives us \( dy = 2dx \) or \( dx = \frac{dy}{2} \).

2. **Change of Limits**: When \( x = 3 \), then \( y = 2 \cdot 3 = 6 \). When \( x = 3 + 3T \), then \( y = 2(3 + 3T) = 6 + 6T \).

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