Explanation

(b)
$\int_{0}^{x^{2}}{f(t)}\ dt = x\cos{\pi x}$
On differentiating both sides, we get
$2x\ f\left( x^{2} \right) = \frac{- x\sin{\pi x}}{\pi} + \cos{\pi x}$
$\Rightarrow \ f\left( x^{2} \right) = - \frac{x\sin{\ \pi x}}{2\pi x} + \frac{\cos{\pi x}}{2x}$
$\therefore\ f(4) = f\left( 2^{2} \right) = \frac{1}{4}$