Why is there only one integration in the solution if there is two integral in the formula? In this problem the random variable is theta and according to the formula there should be two integrations but in the solution there is only one . Nor am i able to understand the meaning of x1 and x2 in the formula. this problem is driving me nuts..
You have a single random variable $\Theta$ in this example, so taking the expectation with respect to that random variable results in a single integral. The formula for $R_X(\tau)$ in your question is rather messy and full of mistakes; it should be as follows:

$$\begin{align}R_X(\tau)&amp;=E[X(t+\tau)X(t)]\\&amp;=E[a\sin(\omega_0t+\omega_0\tau+\Theta)\cdot a\sin(\omega_0t+\Theta)]\\&amp;=\frac{a^2}{2}E[\cos(\omega_0\tau)-\cos(2\omega_0t+\omega_0\tau+2\Theta)]\\&amp;=\frac{a^2}{2}\frac{1}{2\pi}\int_{0}^{2\pi}[\cos(\omega_0\tau)-\cos(2\omega_0t+\omega_0\tau+2\Theta)]d\Theta\\&amp;=\frac{a^2}{2}\cos(\omega_0\tau)\end{align}$$