I would like to learn how to evaluate the following integral. I have checked the table of integrals and could not manage to solve it. Can you please help me?
$\displaystyle\int_{0}^{\infty}\sqrt{\left(1- \frac{1}{(1+x)^2}\right)} e^{ ({-\frac{x}{c}})} dx$
As @Peter Foreman commented, do not expect too much for an antiderivative (even using special functions) or a closed form result.
What we can notice is that $$I(c)=\int_0^\infty\sqrt{1- \frac{1}{(1+x)^2}}e^{ {-\frac{x}{c}}}\,dx <\int_0^\infty e^{ {-\frac{x}{c}}}\,dx=c$$
The only solution left is numerical integration. Trying $$\left( \begin{array}{cc} c & I(c) \\ 1 & 0.743781495 \\ 10 & 9.532405111 \\ 100 & 99.44992339 \\ 1000 & 999.4324691 \\ 10000 & 9999.429709 \\ 100000 & 99999.42916 \end{array} \right)$$ which seems to show that, for large values of $c$ $$I(c) \sim c - 0.57$$ is a rather good approximation.