Prove Integral \int_{0}^{x}\frac{\sin(t)}{t+1}dt > 0 Without Calculus

[Buri]

Homework Statement

Prove that $$\int_{0}^{x}\frac{\sin(t)}{t+1}dt > 0$$

Homework Equations

It asks me to do this without actually calculating it and I can't use the Fundamental Theorem of Calculus.

The Attempt at a Solution

Okay, I see why this should be true as the function oscillates (with smaller bumps over time) around the x-axis so only the first bump really counts. I've tried considering small intervals where I know the function has certain values, but I can't seem to get anywhere with that idea either. Any help?

 

[Buri]

Even a hint would be greatly appreciated! Thanks!

 

[Buri]

Ughhh I figured it out! I just showed that the integral from [0,pi] is larger in absolute value than the integral from [pi,2pi]. Continuing in this fashion (i.e. pairing up consecutive intervals) proves the assertion.