[Math] How to integrate greatest integer function $\int^{1.5}_0 \lfloor x^2 \rfloor \, dx$

calculusfunctionsintegration

How to integrate greatest integer function

$$\int\lfloor x\rfloor \, dx$$

I don't have any idea how to integrate greatest integer function, only have idea about the function viz. If $x = 1.5$ then the value of function will be $2$.

Request you to please elaborate on this, I will be grateful to you.

Thanks in advance.

Best Answer

Hint: Break up into three integrals, (i) $0$ to $1$; (ii) $1$ to $\sqrt{2}$; (iii) the rest. Each will be quick.

To see why this works, plot $\lfloor x^2\rfloor$ in our interval. You will get a staircase pattern.

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