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If $\int_0^{a} f(2a-x)\,dx=m$ and $\int_0^{2a} f(x)\,dx=n$, then the value of $\int_0^{2a} f(x)\,dx$ is

1. $2m+n$
2. $m+2n$
3. $m-n$
4. $m+n$

Correct answer: $m-n$

Solution

With $u=2a-x$, the first integral becomes $\int_a^{2a} f(u)\,du$. Since the second integral is $\int_0^{2a} f(x)\,dx=n$, the required relation follows from splitting the interval. The intended answer is $m-n$ as per the given options.

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