Question: Given $f(x)$ is continuous and $0 \le f(x) \le 1$, and $\int f(x) dx = g(x)$, is there indefinite or definite integration formula for $$\int f(x)^n dx$$ or an approximation or expansion?

Answer: 
You need to show that $\cos^4\theta+\sin^4\theta$ never gets too small. A nice way to do this is to show

$$\begin{align}
\cos^4\theta+\sin^4\theta
&=(\cos^2\theta+\sin^2\theta)^2-2\cos^2\theta\sin^2\theta\\
&=1-{1\over2}(2\sin\theta\cos\theta)^2\\
&=1-{1\over2}\sin^22\theta\\
&\ge1-{1\over2}\\
&={1\over2}
\end{align}$$
To bound the trigonometric part:
$$\cos^4 \theta + \sin^4 \theta = \cdots = \frac{\cos 4t + 3}{4}$$
So this factor lies between $1/2$ and $1$ (See Barry Cipra's answer to derive this lower bound).

It then suffices to show that:
$$\lim\_{r \to 0} \frac{e^{-1/r^2}}{r^4}= 0$$