Evaluating $\int_0^a \lfloor x^n \rfloor \,\mathrm{d}x$ 
Evaluate $\int_0^a \lfloor x^n \rfloor \,\mathrm{d}x$ (where $ \lfloor \,\cdot\, \rfloor $ denotes greatest integer function).

Can anyone please give a detailed explanation of how to do this? 
This is my first question on MathStack Exchange.
Thank You

If you mean $$ \begin{align} 
\int_{0}^{\infty}\lfloor x^n\rfloor dx,
\end{align} $$
then this integral diverges to infinity for all $ n\in\mathbb{R} $. To see this, I suggest making a comparison between $ \int_{0}^{\infty}\lfloor x^n\rfloor dx $ and $ \int_{0}^{\infty} x^n dx $, then using the integral test for convergence.
