Consider the function $f(x) = x^{\cos(x) + \sin(x)}$ defined for $x \geq 0$.
(a) Prove that
$$0.4 \leq \int_{0}^{1} f(x)\, dx \leq 0.5$$
(b) Suppose the graph of $f(x)$ is being traced on a computer screen with the uniform speed of 1 cm per second (i.e., this is how fast the length of the curve is increasing). Show that at the moment the point corresponding to $x = 1$ is being drawn, the $x$ coordinate is increasing at the rate of
$$\frac{1}{\sqrt{2 + \sin(2)}} \text{ cm per second.}$$