> 
> I'm stuck at the following integration problem:
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> a) Given that $x = e^u - e^{-u}$, find an expression for $e^u$ in terms of $x$.
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> b) Use the substitution $x = e^u - e^{-u}$ to find the exact value of $\int\_0^1 \frac {1} {\sqrt {4+x^2}}\, dx$.
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My textbook says that the solutions are:

a) $e^u=\frac {x+\sqrt{x^2+4}}{2}$

b) $\ln\frac {1+\sqrt5}{2}$

I have tried some algebraic manipulation in task a) but this did not bring me far. The textbook solutions looks pretty much like the result of the quadratic formula but I do not know if this is a coincidence or not.

I did not understand b) either since I lack the understanding of task a). I might be able to solve it because you could use the substitution $u =\frac x 2 $ because this nicely leads to $\int\_0^1 \frac 1 {\sqrt{u^2+1}}\,du $ and then I one could trig-substitution but I cannot solve it with the given substitution.

Could somebody tell me I can solve or even approach the problem?