Need help with this integral I=$\int_{0}^{9 \times 10^6} (1-\frac{\lfloor\sqrt x\rfloor}{\sqrt x})$ where $\lfloor x \rfloor$ denotes greatest integer function.
I have tried breaking it into fractional part but couldn't get any good approach and also if i go for breaking the GIF function where ever it is integer…it comes out to be 3001 points in the given range of number in the integration.
Calculus – Integration Including Greater Integer Function
calculusdefinite integralsintegration
Best Answer
For every integer $n$ from $0$ to $3\cdot10^3-1$, $$\int_{n^2}^{(n+1)^2}\left(1-\frac{\lfloor\sqrt x\rfloor}{\sqrt x}\right)dx=\left[x-2n\sqrt x\right]_{n^2}^{(n+1)^2}=1$$ hence $$\int_0^{9\cdot10^6}\left(1-\frac{\lfloor\sqrt x\rfloor}{\sqrt x}\right)dx=3\cdot10^3.$$