Pareto distribution (integration)

Question

The random variable $X$ follows a Pareto distribution with parameters $\alpha$ and $\lambda$.
I need to show the below:

I have come across the following answer, but couldn't understand step 2. I have tried integration by parts with no success.

Answer 1

If your approach to integration by parts is something like $\displaystyle \int_b^c u \, dv=\bigg[uv\bigg]_b^c-\int_b^c v \, du$

then try $u=x$ and $dv=\dfrac{\alpha \lambda^\alpha}{(\lambda+x)^{\alpha+1}}\, dx$, so $du = dx$ and $v=-\dfrac{ \lambda^\alpha}{(\lambda+x)^{\alpha}}$