Chapter 5: Integrals - Section 5.4 - The Fundamental Theorem of Calculus - Exercises 5.4 - Page 288: 68

$1$

The function $f(x)$ is the derivative of $\int_0^{x} f(t) dt=x \cos \pi x$ by the fundamental Theorem of calculus. Here, we have $f(x)=\dfrac{d}{dx} (\int_0^{x} f(t) dt)$ and $\dfrac{d}{dx} (x \cos \pi x)=\cos \pi x- \pi x \sin \pi x$ Thus, $f(4)=\cos (4 \pi)- (4\pi) \sin (4 \pi)=1-0=1$