determine whether the improper integral diverges or converges. evaluate if it converges.

Question

integral of e^-xcosxdx from 0 to infinity

i got the limit as b--> infinity for (-e^-xcosx + e^-xsinx) /2

Would I plug in my value for infinity and subtract it from the value I get when i plug in 0?

Answer 1

that's the short way to do it.
Formally, you should take the limits, but since it is clear that e^-x goes to zero, it's pretty simple.
Even though cos(infinity) and sin(infinity) do not converge to a limit, we know that they are bounded by 1, so e^-x takes over.

Answer 2

so what’s the answer? 1/2?
And my way is still right , right ?