Let $T>0$ be a fixed number. $f: R \rightarrow R$ is a continuous function such that $f(x+T)=f(x), x \in R$.…
Let $T>0$ be a fixed number. $f: R \rightarrow R$ is a continuous function such that $f(x+T)=f(x), x \in R$.
If $I=\int_0^T f(x) d x$, then $\int_0^{5 T} f(2 x) d x=$
$10I$
$\frac{5}{2}$I
$5I$
$2I$
Solution
Given, $I=\int_0^T f(x) d x$
If $f(x+T)=f(x)$
Now, $=\int_0^{5 T} f(2 x) d x$
On putting $2 x=y$
$\Rightarrow \quad d x=\frac{1}{2} d y$
$\frac{1}{2} \int_0^{10 T} f(y) d y=\frac{10 I}{2}=5 I$