Let

$\displaystyle f(x)= \int_{x}^{x+1} sin(e^t) \, dt$

Show that

$\displaystyle e^x|f(x)|<2$

and that

$\displaystyle e^x f(x)=cos(e^x)-\frac{cos(e^{x+1})}{e}+r(x)$

where $\displaystyle |r(x)|<Ce^{-x}$, for some constant C.

I got everything except I don't know where to get the r(x) part at all...