How would you integrate a function containing a definite integral (without calculating the integral)? Suppose you had the function $f(x)=(\int^{x}_{0} e^{-u^{2}} du)^{2}$. How would you find $f'(x)$? The integral in the function is a bit nasty, but I've been told that there is a way to find the derivative without calculating the integral. What method would be used to do this?
Thanks

If we first denote the integral function of the inner function as 
$$ G(x) = \int_0^x e^{-u^2} \ du$$
Now by definition of the integral function we have
$$ G'(x) = e^{-x^2}$$
Now if we consider your function we have
$$ f(x) = (G(x))^2$$
The derivative is found with the chain rule
$$ f'(x) = 2G'(x)G(x)$$
So
$$ f'(x) = 2e^{-x^2} \int_0^x e^{-u^2} \ du$$
