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  4. If ∫ limits0x f(x) sin t d t is constant for every x ∈(0,2 π) and f(π)=2, then the value of f((π/2)) is equal to

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Q. If $\int\limits_0^x f(x) \sin t d t$ is constant for every $x \in(0,2 \pi)$ and $f(\pi)=2$, then the value of $f\left(\frac{\pi}{2}\right)$ is equal to

Integrals

Solution:

$f ( x ) \times \int\limits_0^{ x } \sin t\,dt = C$ (given)
$f(x)(1-\cos x)=C $
$\text { put } x=\pi$
$2(2)=C $
$ \therefore C=4 $
$\Rightarrow f(x)=\frac{4}{1-\cos x} $
$ \Rightarrow f\left(\frac{\pi}{2}\right)=4$