If _ -1 ^1 f(x) d x=0, then - Vidyadip

MCQ

If $\int_{-1}^1 f(x) d x=0$, then

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

✓

Answer

Correct option: B.

$f (-x)=- f (x)$

(B)
$\int_{-1}^1 f (x) d x=\int_{-1}^0 f (x) d x+\int_0^1 f (x) d x$
In $1^{\text {st }}$ integral, put $x=- t \Rightarrow d x=- dt$
$\therefore \quad \int_{-1}^0 f (x) d x=-\int_1^0 f (- t ) dt$
$\begin{array}{l}=\int_0^1 f(-t) d t \\ =\int_0^1 f(-x) d x\end{array}$
$\therefore \quad \int_{-1}^1 f (x) d x=\int_0^1 f (-x) d x+\int_0^1 f (x) d x$
$=0$, if $f (-x)=- f (x)$

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