If _0^ 2 a f(x) d x=2 _0^a f(x) d x, then - Vidyadip

MCQ

If $\int_0^{2 a} f(x) d x=2 \int_0^a f(x) d x$, then

A
$f (2 a -x)=- f (x)$
✓
$f (2 a -x)= f (x)$
C
$f(a-x)=-f(x)$
D
$f ( a -x)= f (x)$

✓

Answer

Correct option: B.

$f (2 a -x)= f (x)$

(B)
$\int_0^{2 a } f (x) d x=\int_0^{ a } f (x) d x+\int_{ a }^{2 a } f (x) d x$
Let $I _1=\int_{ a }^{2 a } f (x) d x$
Put $x=2 a - t \Rightarrow d x=- dt$
$\therefore \quad I_1=-\int_a^0 f(2 a-t) d t$
$=\int_0^a f(2 a-t) d t=\int_0^a f(2 a-x) d x$
$\therefore \quad \int_0^{2 a} f(x) d x=\int_0^a f(x) d x+\int_0^a f(2 a-x) d x$
$=2 \int_0^{ a } f (x) d x, \quad$ if $f (2 a -x)= f (x)$

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