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?

  • 1
    $\begingroup$ Without knowing anything about $F$ or the size of $T$, you are pretty much out of luck. Also, the limits of integration are relevant. That said, you have $e^{-TF} ( 2F - F^2 T)$ as your integrand. This is of the right form to use Laplace's method if you are considering $T \to \infty$. $\endgroup$ – Ian Mar 13 '17 at 1:59

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:

$$ \int_{x_0}^{x_1} (2 - TF(x))F(x)e^{-TF(x)} dx. $$


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