Evaluate $\int_0^\infty \frac{1}{x^{n+2}} c^{1/x} \exp\{-\lambda/x\} \mathrm{dx} $

Question

I have been trying to evaluate the following integral

$$\int_0^\infty \frac{1}{x^{n+2}} c^{1/x} \exp\{-\lambda/x\} \mathrm{dx} $$

What I am getting is

$$\frac{1}{\left(\lambda-logc \right)^{n+1}} \times \Gamma \left( n+1 \right) $$

Could you please check whether this is indeed the correct answer?

Thank you.

Answer 1

The simple substitution $u=\dfrac1x$, coupled with the fact that $c=e^{\ln c}$, should be enough to transform this expression into something more recognizable. Indeed, Mathematica confirms your result.