Variance of function of random variable

Is their an easier way to find variance of function of random variable?

Till now what I am doing is first find probability density function of (function of random variable) then integrate over range.

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