Calculate the distance $h(t)=1+e^{-t}\cos(t)$ that drops A weight is hanging from a spring, and oscillates with heights modeled by the function $h(t)=1+e^{-t}\cos(t)$ at time $t\geq 0.$ Does it travel an infinite distance (in other words, does it converge or diverge) as $t\to\infty$ and also, what is the total distance that it drops ?
Note: the distance dropped indicates only the downward motion of the function, not when the function increases.
I have tried to plot this and it seems from the graph that it only drops to $\frac{3\pi}{4}$ and increases from that point on. I got a final answer of $\frac{1}{1-e^{-\pi}}.$ For the other question of whether the weight travels infinitely, the answer is a simple no.

We have $\cos(t)=\frac{1}{2}\cdot(e^{it}+e^{-it}),$ so rewrite to
$$1+e^{-t}\cdot\frac{1}{2}\cdot(e^{it}+e^{-it})=1+\frac{1}{2}\cdot(e^{t(i-1)}+e^{t(-i-1)})$$
Now integrate over t to get $t+1/2(\frac{e^{t(i-1)}}{i-1}-\frac{e^{t(-i-1)}}{i+1})$ from $0$ to infinity to get infinity and thus it travels an infinite distance.
