Let $T>0$ be a fixed real number. Suppose $f$ is a continuous function such that $f(x+T)=f(x)$ for all $x \in R$. If $I=\int\limits\limits_0^T f(x) d x$ then $I_1=\int\limits_3^{3+3 T} f(2 x) d x$ is
Answer & explanation
Correct answer: option 3
We have, $I=\int\limits\limits_0^T f(x) d x$
Let, $I_1=\int\limits_3^{3+3 T} f(2 x) d x$
$\Rightarrow I_1 =\frac{1}{2} \int\limits_6^{6+6 T} f(t) d t$, where $t=2$
$\Rightarrow I_1=\frac{1}{2} \times 6 \int\limits_0^T f(t) d t $ $\left[∵ \int\limits_a^{a+n T} f(x) d x=n \int\limits_0^T f(x) d x\right]$
$\Rightarrow I_1=3 I$