integral involving greatest integer function 
Let $S_n = \sum_{k=1}^n \frac{1}{k}$ and $I_n=\int_1^n \frac{x-[x]}{x^2}dx$. Then, what is $S_{10} + T_{10}$?

The only clue that i can get is break the limits of integration according as the value the greatest integer function takes???please solve this for me. The answer i got was $\ln10 + 1$.

Hint 1:

 \begin{align}\int_1^n \frac{x-[x]}{x^2}dx &= \int_1^2 \frac{x-1}{x^2} dx+\int_2^3\frac{x-2}{x^2} dx + \cdots + \int_{n-1}^n\frac{x-n+1}{x^2} dx\end{align}

Hint 2:

 \begin{align}\int_{n-1}^n \frac{n-1}{x^2}dx&=\left[\frac{(1-n)}{x}\right]_{n-1}^n\\&=(1-n)\left(\frac{1}{n}-\frac{1}{n-1}\right)\\&=\frac{1}{n}\end{align}

Can you proceed from it?
