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If $f(x)=f(a-x)$ then $\int_0^a x f(x) d x$ is equal to
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Verified Answer
The correct answer is:
$\frac{a}{2} \int_0^a f(x) d x$
Hints : $f(x)=f(a-x), I=\int_0^a x f(x) d x=\int_0^a(a-x) f(a-x) d x$
$$
\begin{aligned}
& =\int_0^a(a-x) f(x) d x=a \int_0^a f(x) d x-I \\
& \therefore 2 I=a \int_0^a f(x) d x \Rightarrow I=\frac{a}{2} \int_0^a f(x) d x
\end{aligned}
$$
$$
\begin{aligned}
& =\int_0^a(a-x) f(x) d x=a \int_0^a f(x) d x-I \\
& \therefore 2 I=a \int_0^a f(x) d x \Rightarrow I=\frac{a}{2} \int_0^a f(x) d x
\end{aligned}
$$
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