Let $f$ be a bounded and with a continuous derivative at the interval $[a,\infty)$.
The integral: $\displaystyle \int_a^\infty f(x) \, dx$ diverges.
Also: $$ \exists t> a, \forall x>t: f'(x) < f(x) $$
Prove that the $\displaystyle \int_a^\infty f(x) \sin(e^x) \, dx$ conditionally converges.
What i tried:
So i want to show that it diverges in its absolute value and converges in its "normal" value.
For diverges in its absolute value
We know that: $$ |\sin(e^x)| < 1 $$
Therefore we can write: $$ \int_a^\infty |f(x)| \leq \int_a^\infty |f(x)\sin(e^x)| \, dx $$
But we know that the left integral diverges from the question, therefore, by comparison test: $$ \int_a^\infty |f(x)\sin(e^x)| \, dx $$ diverges.
Now the problem in proving converges for the "normal" function.
I thought to use Dirichlet test, but i dont see how to say that $f(x)$ is decreasing monotonic or to talk about the limit.
I must say it very sounds like drichlet test, but i cant see how it feets...
Couldnt think of other functions for dirichle
So i thought about the comparison test, yet couldn't think of a converging function that will fit.
In the end, i am stuck.
Those are my homework, so i prefer a hint than a solution.