Calculate the integral $\int_0^{x} (\lfloor t+1 \rfloor)^3dt$

Question

Here's the integral, $$\int_0^{x} (\lfloor t+1 \rfloor)^3dt$$

I have some knowledge of computing the integrals of discontinous functions but the cube function and the independent variable limit confused me a bit.
Also note the cube is outside the floor function as written.
Thanks for any help..

Split the interval of integration into multiple parts at strategically chosen points. — Daniel Fischer, Jul 29 '13 at 18:44
The integrand is complicated in the whole line but simple in the intervals with integer endpoints. — lhf, Jul 29 '13 at 18:47
@Daneil for example? can you give me first two splits please? — Simar, Jul 29 '13 at 18:47

Jakube · Accepted Answer · 2013-07-29T19:00:42.970

5

$$\int_0^x [ t+1 ]^3 dt = \sum_{i=1}^{[x]}\int_{i-1}^{i}[ t+1 ]^3 dt + \int_{[x]}^{x}[ t+1 ]^3 dt $$ $$=\sum_{i=1}^{[x]}\int_{i-1}^{i}i^3 dt + \int_{[x]}^{x}[x+1]^3 dt =\sum_{i=1}^{[x]}i^3+ [x+1]^3\cdot(x-[x]) $$ $$ = \frac{[x]^4+2[x]^3+[x]^2}{4} + [x+1]^3\cdot(x-[x])$$

edited Jul 29 '13 at 19:00

answered Jul 29 '13 at 18:49

Jakube

1,895

3

What if $x$ isn't an integer? – Daniel Fischer Jul 29 '13 at 18:50
The last term of sum should be different in case of non-integer $x$, like $(x-\lfloor x \rfloor)\cdot (\lfloor x \rfloor)^3$ – Evgeny Jul 29 '13 at 18:50
I corrected it. – Jakube Jul 29 '13 at 18:55
2

You have a small mistake, it's $$\int_{\lfloor x\rfloor}^x (\lfloor x \rfloor + 1)^3, dt$$ (and consequently the last summand is $(\lfloor x\rfloor + 1)^3\cdot (x - \lfloor x\rfloor)$. But in principle it's fine now. – Daniel Fischer Jul 29 '13 at 18:58
The integrand is ${\lfloor 1+t \rfloor}^3 $ not $[1+t]^3$. – Mhenni Benghorbal Jul 29 '13 at 19:31
Got it all Thanks @DanielFischer and user38034 :) – Simar Jul 29 '13 at 19:59
So does this method work for all discontinous integrals ? – Simar Jul 29 '13 at 20:00
@Simar You can always split at the discontinuities. But of course not always there is such a nice sum for the sum of the integrals over the intervals of continuity. – Daniel Fischer Jul 29 '13 at 20:12

Calculate the integral $\int_0^{x} (\lfloor t+1 \rfloor)^3dt$

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