Let T > 0 be a fixed real number. Suppose f is a continuous function such that for all x ∈ R f(x+T)= f(x)…

Q: Let T > 0 be a fixed real number. Suppose f is a continuous function such that for all x ∈ R f(x+T)= f(x). If $I = \int_{0}^{T} f(x) dx $ , then the value of $ \int_{0}^{3+3T} f(2x) dx $  is

(A) $\frac{3}{2}I $

(B) 2I

(C) 3I

(D) 6I

Sol: Let $\large L = \int_{0}^{3 + 3T} f(2x) dx $

Put 2x = t so that

$\large L = \frac{1}{2}\int_{0}^{6 + 6T} f(t) dt $

$\large L = \frac{6}{2}\int_{0}^{T} f(t) dt = 3 I$

Hence (C) is the correct answer.