Variance of function of random variable
Assume $Y = g(X)$ be the function of RV $X$, then by using the following

$$E\{ g(X) \} = \int g(x) f_X(x) dx $$

variance of $Y$ can be computed without the computation of pdf $f_Y(y)$ as:

$$
\begin{align}
\text{Var(Y)} &amp;= E\{ (Y-\mu_Y)^2 \} = E\{ Y^2 \} - (\mu_Y)^2 \\
&amp; =  E\{ g^2(X) \} - E\{ g(X) \}^2 \\
&amp; = \int g^2(x) f_x(x) dx  - \left(\int g(x) f_X(x) dx \right)^2 \\
\end{align}
$$