I have problem of solving or approximating the following integral:
$$ \int_{0}^{\infty} \frac{(1- (ax+1)^{(1-n)})^{(m-1)}}{(ax+1)^n} dx$$
I tried substitution or simplification, but it did not work. It was not successful.
Can anyone suggest any tips please?
Thank you.
I am assuming $a>0,n>1$.
Keep $1-(ax+1)^{1-n}=u,du=a(n-1)dx/(ax+1)^n$ which gives$$I=\int_{0}^1\frac{u^{m-1}}{a(n-1)} du=\begin{cases}\frac1{am(n-1)},&m>0\\\infty,&m\le0\end{cases}$$
For $n=1$ the integrand is $0$. For $n<1$ we get $-\infty$ for $m\le0$ but the same result as above for $m>0$.