How can I prove that f(x) is greater than 0 for x≥0?

[monsmatglad]

Homework Statement

Hi. I need help understanding a task where i am supposed to prove that a function must be greater than 0 when x is from 0 and up. f(x) = (x-0)integral of (sinx/(x+1)) please help me out with this.

Mons

 

[Tedjn]

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.

 

[monsmatglad]

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.

sorry, but i don't seem to understand.:/ what do you mean with breaking the integral into sub-integrals. could you please give me an example.

Mons

 

[Tedjn]

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.

 

[monsmatglad]

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.

I've been looking at the function and at the x-values for maximum and minimum, but i don't see how to get any information about the function(the integral) except for its derivative. if i split up in sub-intervals i don't understand find anything about the integral since it can't be analytically solved and therefor can't produce any integral values.

Mons
Sorry for bad English.

 

[Tedjn]

As in most inequalities, you don't need to find an exact solution, only find a way to convincingly show a sum is positive. There may be many ways to do this, but you can try to combine the sub-integrals.