# Closed form solution to $\int_0^{\infty} x(1+({\frac{x}{\lambda}})^{(-\alpha-1)}) dx$

Is there a closed form solution for this integral?

$${\int_0^{\infty} x(1+{\frac{x}{\lambda}})^{(-\alpha-1)} dx}$$

where $$\alpha>1$$ $$\lambda>0$$

It's in Pareto statistical distribution family and I'm trying to find an analytical solution.

• Does the$\int_{0}^{\infty} xdx$ converge? Sep 20, 2019 at 20:08
• Good point! I forgot the top-half of the equation. I've edited it now Sep 20, 2019 at 20:12
• Thanks for pointing that out. It's fixed now Sep 20, 2019 at 20:15
• Welcome, You may edit the title of the equation also ) ; Sep 20, 2019 at 20:26

The integral in the denominator looks like the expected value of a Lomax distribution. (some constants missing) I read that you need $$\alpha >1$$, otherwise undefined. Looking at the Wikipedia material, It appears the solution is
$$\int_0^{\infty} x\left(1+{\frac{x}{\lambda}}\right)^{(-\alpha-1)}dx = \frac{\lambda^2}{\alpha(\alpha -1)}$$
$$I=\int_{0}^{\infty} \frac{x}{(1+x/b)^{(1+a)}} dx =b^{(1+a)} \int_{0}^{\infty} \left( \frac{1}{(b+x)^a} - \frac{b}{(b+x)^{1+a}} \right)=b^{1+a} \left( \frac{(b+x)^{1-a}}{1-a}+b \frac{(b+x)^{-a}}{a} \right)_{0}^{\infty}=\frac{b^2}{a(a-1)},~ a > 1$$