_0^1 f(1 - x) ,dx has the same value as the integral - Vidyadip

MCQ

$\int_0^1 {f(1 - x)\,dx} $ has the same value as the integral

✓
$\int_0^1 {f(x)\,dx} $
B
$\int_0^1 {f( - x)\,dx} $
C
$\int_0^1 {f(x - 1)\,dx} $
D
$\int_{ - 1}^1 {f(x)\,dx} $

✓

Answer

Correct option: A.

$\int_0^1 {f(x)\,dx} $

a
(a) Put $1 - x = t \Rightarrow - dx = dt$.

Also as $x = 0$ to $1,$ $t = 1$ to $0$

Therefore, $\int_0^1 {f(1 - x)dx = \int_1^0 {f(t)( - dt)} } = \int_0^1 {f(t)dt = \int_0^1 {f(x)dx} } $.

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