Question

Let $T>0$ be a fixed number. Suppose f is a continuous function such that for all $xϵR,f(x+T)=f\left(x\right).$ If $I={\int }_0^{T}f\left(x\right)dx$ then the value of ${\int }_3^{3+3T}f\left(2x\right)dx$ is

• A. $\frac{3}{2}I$
• B. $2I$
• C. $3I$
• D. $6I$

Answer 1

The correct option is C $3I$

Let $J={\int }_3^{3+3I}f\left(2x\right)dx$
Substituting $2x=t\Rightarrow dx=\frac{dt}{2}$, we get
$J=\frac{1}{2}{\int }_6^{6(1+T)}f\left(t\right)dt=\frac{6}{2}{\int }_0^{T}f\left(t\right)dt=3{\int }_0^{T}f\left(x\right)dx=3I$