Numerical integration problem

Question

How can I solve integral equations of the form $$\int_{-3}^x e^{e^t}dt=3?$$ Is there for example Sage code for that kind of equations? Is there better method that evaluating numerically

$$\int_{-3}^{-1} e^{e^t}dt<2.4,$$

$$\int_{-3}^{0} e^{e^t}dt>4.2,$$

$$\int_{-3}^{-0.5} e^{e^t}dt>3.1,$$

and so on?

Answer 1

Using the nice hint by @HaraldHanche-Olsen, we can cast this into an ODE problem and use a numerical solver.

So, we have:

$$y' = e^{e^x},~~ y(-3) = 0$$

Using a fourth order Runge-Kutta, with a step-size of $0.045$, we get the following values:

• $\ldots$
• $x = -0.615, y = 2.95805$
• $x = -0.57, y = 3.03628$
• $x = -0.525, y = 3.11653$

If we numerically evaluate the integral at the point we found above, we get an exact match.

Now that you have that range, you can use various methods to reduce the error to whatever you need.