How to solve $\int_{0}^{\infty} e^{-ax} \frac{x}{\sqrt{1+x^2}}dx$ or $\int_{0}^{\infty} e^{-ax} \sqrt{1+x^2}dx$

Question

I need to solve either of the following integrals, they are related to each other by integration by parts:

$$I_1 = \int_{0}^{\infty} e^{-ax} \frac{x}{\sqrt{1+x^2}}dx$$

or

$$I_2 = \int_{0}^{\infty} e^{-ax} \sqrt{1+x^2}dx$$

I have not been able to find them in integral tables and I wonder if it is possible to solve either of these integrals.

Any ideas on how to calculate $I_1$ or $I_2$?

Answer 1

If you notice it, you will see that this is the form of Laplace transformation. Thus the first integration can be written as the Laplace transformation of (x/root(x^2 +1)) The second integral can be written by integration by parts Which is e^-ax * root(x^2 +1) - integrate(-ae^-at * root(x^2 +1) dx) Which is then equals to e^-ax * root(x^2 +1) + a*laplace of (root(x^2 +1))

Hope this helped