The discussion revolves around proving that a function, defined as f(x) = (x-0)∫(sin(x)/(x+1))dx, is greater than 0 for x ≥ 0. Participants are exploring the behavior of the integral and the function within the specified domain.
Some participants have offered guidance on applying properties of integrals to explore the problem further. There is an acknowledgment of the difficulty in finding exact solutions, with suggestions to focus on demonstrating that a sum of integrals is positive rather than seeking an explicit solution.
Participants express uncertainty about the function's behavior and the challenges posed by the integral's inability to be solved analytically. There is also a concern about maintaining a balance between providing help and avoiding complete solutions.
Tedjn said:Here is the method with which I imagined doing this. Find when sin(x)/(x+1) changes sign. Break the integral from 0 to x into many sub-integrals, from 0 to first sign change, from first sign change to second sign change, etc. See if you can combine some integrals together to get an equivalent sum of all positive numbers. Ergo, the final integral is positive.
EDIT: I corrected the part in bold above, upon rereading this.
Tedjn said:Repeatedly apply, if a < c < b,
\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx
I am sorry if I am too vague, but it is a fine line between helping and solving. Apply this equation to what I said above, at the points I said above, and see if inspiration strikes. After expanding, you can try combining integrals together again, but in a different way than the way you broke them up. Note that you can carefully change the bounds on an integral as long as you make the appropriate compensations inside the body of the integral.