Integral of a line with random gradient

Question

Consider a random line $Y = Mx$ where $M$ is a standard normal variable $M \sim \mathcal{N}(0,1)$. The line is integrated between 0 and 1:

$$I = \int_{0}^{1} Y dx = \int_{0}^{1} Mx dx$$

What is the probability density function of the integral $I$?

Answer 1

$M$ and $x$ are independent? $\int_0^1Mx=M\int_0^1x=M/2$ so $I\sim N(0,1/4)$.