Question

If $\underset{0}{\overset{x}{\int }}f(x)\sin tdt$ 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

• A. 4
• B. 2
• C. $\frac{1}{2}$
• D. $\frac{\pi }{2}$

Answer 1

Answer: (A)

$f(x)\times \underset{0}{\overset{x}{\int }}\sin tdt=C$ (given)
$f(x)(1-\cos x)=C$
$\text{ put }x=\pi$
$2(2)=C$
$\therefore C=4$
$\Rightarrow f(x)=\frac{4}{1-\cos x}$
$\Rightarrow f\left(\frac{\pi }{2}\right)=4$