Question

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

• A. $\int\limits^5_{x+5} g (t)\, dt$
• B. $5 \int\limits^5_{x+5} g (t) \,dt$
• C. $\int\limits^{x+5}_5 g (t) \,dt$
• D. $2 \int \limits ^{x+5}_5 g (t) \,dt$

Answer 1

Answer: (A)

$\int\limits^5_{x+5} g (t)\, dt$

Answer 2

$f( x )=\int_{0}^{ x } g ( t ) dt$
$f(- x )=\int_{0}^{- x } g ( t ) dt$
put $t =- u$
$=-\int_{0}^{ x } g (- u ) du$
$=-\int_{0}^{x} g(u) d(u)=-f(x)$
$\Rightarrow f(- x )=-f( x )$
$\Rightarrow f( x )$ is an odd function
Also $f(5+x)=g(x)$