# Expected value of joint density

How should I approach this problem?

The joint density for $(X,Y)$, where $X$ is the arrival time of the first vehicle from the north-south direction and $Y$ is the arrival time of the first vehicle from the east-west direction at an intersection, is given by

$$f(x)=\frac{1}{x}$$ $$0<y<x<1$$

Find $E[X]$.

Before this I've only encountered expected value-problems where I've either been given a table to find direct values.

Following the formula for expected values via joint density:

$$\int _{-\infty }^{\infty }\:\int _{-\infty }^{\infty }\:xf_{XY}\left(x,y\right)dxdy$$

I try something like this but I don't get the right answer, it's supposed to be $\frac{1}{2}$.

$$\int _0^x\:\int _y^1\:x\frac{1}{x}dxdy$$

I might be far off with this but I have no idea. Any help would be very appreciated!