Search any question & find its solution
Question:
Answered & Verified by Expert
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 $\int_0^{5 T} f(2 x) d x=$
Options:
If $I=\int_0^T f(x) d x$, then $\int_0^{5 T} f(2 x) d x=$
Solution:
1963 Upvotes
Verified Answer
The correct answer is:
$5I$
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 $2 x=y$
$\Rightarrow \quad d x=\frac{1}{2} d y$
$\frac{1}{2} \int_0^{10 T} f(y) d y=\frac{10 I}{2}=5 I$
If $f(x+T)=f(x)$
Now, $=\int_0^{5 T} f(2 x) d x$
On putting $2 x=y$
$\Rightarrow \quad d x=\frac{1}{2} d y$
$\frac{1}{2} \int_0^{10 T} f(y) d y=\frac{10 I}{2}=5 I$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.