Question

If $f\left(x\right)$ is a continuous function and ${\int }_{x^2}^{x^4}{t}^{3}f\left(t\right)dt=sin2\pi x$ , 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,
$x^{12}f\left(x^4\right)4x^3-x^{6}f(x^2)2x=2\pi cos\pi x$
Putting $x=1$
$\Rightarrow 4f\left(1\right)-2f\left(1\right)=2\pi$
$\Rightarrow 2f\left(1\right)=2\pi \Rightarrow f\left(1\right)=\pi$