Let f ( x ) = _0^x g ( t )dt , where g is a non zero even functio… - Vidyadip

MCQ

Let $f\left( x \right) = \int\limits_0^x {g\left( t \right)dt} $, where $g$ is a non zero even function. If $f(x+5) = g(x)$ , then $\int\limits_0^x {f\left( t \right)dt} $ equals

✓
$\int\limits_{x + 5}^5 {g\left( t \right)dt} $
B
$2\int\limits_{5}^{x - 5} {g\left( t \right)dt} $
C
$\int\limits_{5}^{x + 5} {g\left( t \right)dt} $
D
$5\int\limits_{x + 5}^5 {g\left( t \right)dt} $

✓

Answer

Correct option: A.

$\int\limits_{x + 5}^5 {g\left( t \right)dt} $

a
since $g(x)$ is even with $f(0)=0$

$f(x)$ is odd function

$g(x)=f(x+5)$

$g(-x)=f(-x+5)$

$g(x)=-f(x-5)$

Replace $x$ by $x+5$ $\Rightarrow f(x)=-g(x+5)$

$\int_{0}^{x} f(t) d t=-\int_{0}^{x} g(t+5) d t$

$=-\int_{5}^{x+5} g(t) d t$

$=\int_{x+5}^{5} g(t) d t$

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