Mathematics

# If $f(x)=\displaystyle\int^x_0(\cos (\sin t)+\cos(\cos t)dt$, then $f(x+\pi)$ is?

$=f(\pi)+2f(\dfrac{\pi}{2})$

##### SOLUTION
$f(x)+f(\pi)$
$f(x+\pi)=\displaystyle\int^{x+\pi}_0(\cos(\sin t)+\cos(\cos t))dt$
$=\displaystyle\int^{\pi}_0(\cos(\sin t)+\cos (\cos t))dt$
$+\displaystyle\int^{x+\pi}_{\pi}(\cos (\sin t)+\cos (\cos t))dt$
$=f(\pi)+\displaystyle\int^x_0(\cos (\sin t)+\cos(\cos t))dt$
$[\therefore$ for $g(x)=\cos(\sin x)+\cos(\cos x), f(x+\pi)=f(x)]$
$=f(\pi)+f(x)$
$=f(\pi)+2f(\dfrac{\pi}{2})$
$[\therefore g(x)$ has period $\pi/2]$.

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

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