_0^a f(x) d x= - Vidyadip

MCQ

$\int_0^a f(x) d x=$

A
$\int_0^a f(a+x) d x$
B
$\int_0^{ a } f (2 a +x) dx$
C
$\int_0^a f(x-a) d x$
✓
$\int_0^{ a } f ( a -x) dx$

✓

Answer

Correct option: D.

$\int_0^{ a } f ( a -x) dx$

(D)
Put $x= a - t \Rightarrow d x=- dt$
When $x=0, t = a$ and when $x= a , t =0$
$\therefore \quad \int_0^{ a } f (x) d x=-\int_{ a }^0 f ( a - t ) dt$
$=\int_0^{ a } f ( a - t ) dt \quad \ldots \left[\because \int_{ a }^{ b } f (x) d x=-\int_{ b }^{ a } f (x) d x\right]$
$=\int_0^{ a } f ( a -x) d x \quad \ldots \left[\because \int_{ a }^{ b } f (x) d x=\int_{ a }^{ b } f ( t ) dt \right]$

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