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

$$\begin{align}
\cos^4\theta+\sin^4\theta
&=(\cos^2\theta+\sin^2\theta)^2-2\cos^2\theta\sin^2\theta\\
&=1-{1\over2}(2\sin\theta\cos\theta)^2\\
&=1-{1\over2}\sin^22\theta\\
&\ge1-{1\over2}\\
&={1\over2}
\end{align}$$
Another way of getting
Barry Cipra's result:

$c^4+s^4
=c^4+(1-c^2)^2
=c^4+1-2c^2+c^4
=1+2c^4-2c^2
=1-2c^2(1-c^2)
\ge 1/2
$
since
$x(1-x) \le 1/4$
for $0\le x \le 1$.