$Z$ ~ Norm(0, 1) Find m.g.t of $x$

Question

$Z$ ~ Norm(0, 1) Find m.g.t of $x$

$m_z(t) = E[e^{tz}] = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} e^{-\frac{1}{2}z^2 + tz}dz = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} e^{z(-0.5z + t)}dz$

not sure how to proceed.

Answer 1

$$\int_{-\infty}^{\infty} e^{-\frac12z^2+tz-\frac12t^2+\frac12t^2}dz=e^{\frac12t^2}\int_{-\infty}^{\infty} e^{-\frac12(z-t)^2}dz=e^{\frac12t^2}\int_{-\infty}^{\infty} e^{-\frac12z^2}dz=e^{\frac12t^2}\sqrt{2\pi}$$