Question

If $f\left(x\right)$ is a continuous function and $\displaystyle \int _{x^{2}}^{x^{4}} t^{3} f \left(t\right) d t=sin2\pi x$ , then $f\left(1\right)$ is equal to

• A. $1$
• B. $-1$
• C. $\pi $
• D. $-\pi $

Answer 1

Answer: (C)

Differentiating using Leibnitz rule, we get,
$x^{12}f\left(x^{4}\right)4x^{3}-x^{6}f\left(\right.x^{2}\left.\right)2x=2\pi cos\pi x$
Putting $x=1$
$\Rightarrow 4f\left(1\right)-2f\left(1\right)=2\pi $
$\Rightarrow 2f\left(1\right)=2\pi \Rightarrow f\left(1\right)=\pi $