Let $n$ is a natural number. Find $\int_0^n 2x \lfloor x \rfloor dx$

Question

Actually, I know that $\int_0^n \lfloor t \rfloor dt$ is $\frac{n(n-1)}{2}$. But I can't solve this simple problem. Can anybody help me? P.S. I'm seventeen, I'm studying in last year of high school, and I'm preparing for the national exams.

Answer 1

\begin{eqnarray*} \int_0^n 2x \lfloor x \rfloor dx = \sum_{i=0}^{n-1} i \underbrace{\int_{i}^{i+1} 2x \,dx}_{x^2 \mid_i^{i+1}=2i+1} = \underbrace{\sum_{i=0}^{n-1} i (2i+1)}_{\text{using} \sum_{i=0}^{n-1} i^2=\frac{n(n-1)(2n-1)}{6} } = \frac{(n-1)n(n+1)}{3}. \end{eqnarray*}

Answer 2

hint: $\displaystyle \int_{0}^n = \displaystyle \int_{0}^1 +\displaystyle \int_{1}^2 +...+\displaystyle \int_{n-1}^n$