Question

If $f\left(x\right)={\int }_0^x\left(cos\right(sint)+cos(cost\left)\right)dt$, then $f(x+\pi )$ is

• A. $f\left(x\right)+f\left(\pi \right)$
• B. $f\left(x\right)+2f\left(\pi \right)$
• C. $f\left(x\right)+f\left(\frac{\pi }{2}\right)$
• D. $f\left(x\right)+2f\left(\frac{\pi }{2}\right)$

Answer 1

Answer: (A), (D)

$f\left(x\right)+2f\left(\frac{\pi }{2}\right)$
$f\left(x\right)+f\left(\pi \right)$

$f(x+\pi )={\int }_0^{x+\pi }\left(cos\right(sint)+cos(cost\left)\right)dt$
$={\int }_0^{\pi }\left(cos\right(sint)+cos(cost\left)\right)dt$
$+{\int }_{\pi }^{x+\pi }\left(cos\right(sint)+cos(cost\left)\right)dt$
$=f\left(\pi \right)+{\int }_0^x\left(cos\right(sint)+cos(cost\left)\right)dt$
$[\therefore$ for $g\left(x\right)=cos\left(sinx\right)+cos\left(cosx\right),f(x+\pi )=f\left(x\right)]$
$=f\left(\pi \right)+f\left(x\right)$
$=f\left(\pi \right)+2f\left(\frac{\pi }{2}\right)$
$[\therefore g(x)$ has period $\pi /2]$
Ans: A,D