Question: I'm doing a physics problem and it boils down to finding this assuming my work is correct:

$$\int\frac{\sqrt{(g/k)}(e^{2t\sqrt{g/k}}-1)}{1+e^{2t\sqrt{g/k}}}~dt$$ to me this looks absolutely horrible to evaluate and was wondering if anyone here could help? Thanks.

$g,k$ are constants.

Answer: 
A closed subset of a complete space is complete. So if $X$ is complete, its unit ball is too.

On the other hand, if the unit ball is complete and $\{x\_k\}$ is a Cauchy sequence in $X$, there must exist a constant $M$ such that $\|x\_k\| \le M$ for all $k$: Cauchy sequences are bounded. Let $y\_k = \frac{x\_k}{M}$ so that $\{y\_k\}$ is Cauchy and $\|y\_k\| \le 1$ for all $k$. Thus $y\_k \to y$ in the unit ball, so that $x\_k \to My$.
**Hints:**

"$\Rightarrow$:"

1. Let $(x\_k)\_{k \in \mathbb{N}}$ be a Cauchy sequence in the closed unit ball $B[0,1]$. Show that there exists $x \in X$ such that $x\_k \to x$.
2. Conclude from $\|x\_k\| \leq 1$ that $\|x\| \leq 1$.

"$\Leftarrow$:"

1. Let $(x\_k)\_{k \in \mathbb{N}}$ be a Cauchy sequence in $X$. Show that there exists a constant $M>0$ such that $$\|x\_k\| \leq M \qquad \text{for all} \, k \in \mathbb{N}.$$
2. Show that the sequence $\tilde{x}\_k := x\_k/M$ is a Cauchy sequence in $B[0,1]$.
3. Conclude.