Let $1>\alpha >0$. Compute $\int_{0}^{\alpha} \exp(\sigma \Phi^{-1}(1-\gamma))d\gamma$ where $\Phi$ is the cumulative distribution of a standard normal distributed Random Variable.
My idea:
$\int_{0}^{\alpha} \exp(\sigma \Phi^{-1}(1-\gamma))d\gamma$. set $ \Phi(k) = (1-\gamma)$ and hence $k = \Phi^{-1}(1-\gamma)$ while $\Phi^{'}(k)dk=-d\gamma$ and hence:
$\exp(-\frac{k^{2}}{2})dk=-d\gamma$ and thus:
$\int_{0}^{\alpha} \exp(\sigma \Phi^{-1}(1-\gamma))d\gamma=\int_{\infty}^{\Phi^{-1}(1-\alpha)} \exp(\sigma k)\times (-\exp (-\frac{k^{2}}{2}))dk$
How do I proceed from here?