1. ## integral/sin proof

prove that the integral from 0 to x of the function [sin(t)]/(t+1) with respect to t is greater than 0 for all x>0

i dont know how to do the math script but it looks something like this
subscript (0) superscript (x) integral([sin(t)]/(t+1))dt >0 for all x>0

2. More generally: consider $f$ a continuously differentiable function, positive and decreasing for $(0,\infty)$ then $\int_0^x f(t)\sin t\,dt>0.$

Proof: integrate by parts and get $\int_{0}^{x}{f(t)\sin t\,dt}=\bigg. (1-\cos t)f(t) \bigg|_{0}^{x}-\int_{0}^{x}{(1-\cos t)f'(t)\,dt}.$

Now $1-\cos t>0$ and $f$ was given as positive, hence $(1-\cos x)f(x)>0,$ on the other hand $f$ was given as decreasing, so $f'(t)<0,$ thus the whole RHS is positive, as claimed. $\blacksquare$