I have a question about how to calculate/approximate an integral:
I have the following integral:
$$ \int_{{x_0}}^{{x_1}} dx \frac{\frac{d}{dx} \left (F^{2}(x) e^{-T F(x)} \right )}{\frac{dF}{dx}} $$
Where $F(x)$ is an arbitrary function and T is a constant function, is there a way to calculate/approximate this integrals? Or is under what conditions of F can I neglect the derivative in the denominator and use the fundamental theorem of calculus?
My guess is this: If $F(x)$ is continuous and differentiable at all $x$, then we can say
$$ \frac{\frac{d}{dx}(F^2(x)e^{-TF(x)})}{\frac{dF}{dx}} = \frac{d}{dF}{(F^2e^{-TF})}\\ = (2F(x) - TF^2(x))e^{-TF(x)}. $$
So the integral would become:
$$ \int_{x_0}^{x_1} (2 - TF(x))F(x)e^{-TF(x)} dx. $$