$\int_0^3 (x^2 + 1){\rm d}\left[x\right]$ where [.] is gif

Question

The question requires us to find $\int_0^3 (x^2 + 1){\rm d}\left[x\right]$ where [.] denotes greatest integer function

My attempt: For... $$\left[x\right] =0\text { , } x\in[0,1) \text { , } x^2+1 \in [1,2)\\ \left[x\right] =1\text { , } x\in[1,2) \text { , } x^2+1 \in [2,5)\\ \left[x\right] =2\text { , } x\in[2,3) \text { , } x^2+1 \in [5,10)\\$$

Plotting them would look like

The shaded area sums up to 9. So I thought it would give ans 9.

But the original answer key says 17. On onservation, it is visible as the sum of shaded as well as non-shaded area just under it.

Could anyone suggest a method to proceed?

Does this answer your question? Interesting Integration with respect to $[x]$ where $[\cdot]$ is greatest integer function or Evalute the definite integral: $\int_{0}^{3} (x^2+1) d[x]$ or Integration involving $d[x]$ instead of dx. — projectilemotion, Jun 14 '21 at 20:17

score 1 · Answer 1 · answered Jun 14 '21 at 20:07

1

You can use "integration by parts" to verify

$$\int_0^3 (x^2+1)d[x] = [x](x^2+1)\Bigr|_0^3 - \int_0^3 2x[x]\:dx = 30 - 10 - 3 = 17$$

answered Jun 14 '21 at 20:07

Ninad Munshi

34,407

$\int_0^3 (x^2 + 1){\rm d}\left[x\right]$ where [.] is gif

1 Answers1