Approximating an integral in the from $e^{1/(a+bx/c)}$

Question

I want to integrate the equation $e^{1/(a+bx/c)}$ with respect to x, where a, b, c are all constants, but I can't seem to find an approximation for it. Is there a way to approximate this integral or the equation itself?

Answer 1

First of all, Welcome to the site !

Let $k=\frac b c$ $$I=\int e^{\frac{1}{a+k x}}\,dx$$ Now $$\frac{1}{a+k x}=t \implies x=\frac{1-a t}{k t}\implies dx=-\frac{dt}{k t^2}\implies I=-\frac 1 k\int \frac{e^t}{t^2}\,dt$$ Use one integration by parts and think about the exponential integral function.