How can [x] be moved into the integrand in the ∫ x^2 d[x] equation?

kushan
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I was going through a book
Which had this question
∫ x^2 d[x] from -5 to 7

Which means integration wrt to greatest integer function of x
Any idea on how to go about it
 
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kushan said:
I was going through a book
Which had this question
∫ x^2 d[x] from -5 to 7

Which means integration wrt to greatest integer function of x
Any idea on how to go about it

d[x]=Ʃ_n δ(x-n) dx as far as I can see it ... or maybe integration by parts is less ambiguous
 
Last edited:
Can you please explain more
 
∫xdy=xy-∫ydx, you can move [x] into the integrand
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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