Calculating Variance when there is a random variable in the exponent

Question

I'm struggling with calculating $$V(e^U+e^{1-U})\qquad\text{while}\qquad U\sim U(0,1)$$ My problem is that I don't know how to approach exercises where there is a random variable in the exponents.
I tried to play with the expectation and with the definition but i can't advance.
Any clues guys?

Answer 1

Let $Z=e^U+e^{1-U}$. Then $\text{Var}(Z)$ is computed as follows: \begin{align*} \text{Var}(Z)&=\mathbb{E}(Z^2)-\mathbb{E}(Z)^2\\ &=\int_0^1z^2\mathrm{d}u-\left(\int_0^1z\mathrm{d}u\right)^2\\ &=\int_0^1e^{2u}+2e+e^{2-2u}\mathrm{d}u-\left(\int_0^1e^u+e^{1-u}\mathrm{d}u\right)^2 \end{align*} Can you compute these integrals?