determine whether the improper integral diverges or converges. evaluate if it converges.
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?
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.
so what’s the answer? 1/2?
And my way is still right , right ?
To determine whether the improper integral converges or diverges, you need to evaluate the limit of the integral as the upper limit approaches infinity.
To find the limit as b approaches infinity for the integral of e^(-x)cos(x) dx from 0 to b, you correctly computed:
(-e^(-b)cos(b) + e^(-b)sin(b)) / 2
Now, to determine if the improper integral converges, you need to evaluate the limit as b goes to infinity.
When you plug in infinity into the expression, you have:
(-e^(-infinity)cos(infinity) + e^(-infinity)sin(infinity)) / 2
As you evaluate these terms, you'll find that e^(-infinity) approaches 0, cos(infinity) and sin(infinity) are oscillating between -1 and 1 without settling to a specific value. Therefore, (-e^(-infinity)cos(infinity) + e^(-infinity)sin(infinity)) / 2 does not have a well-defined limit.
Since the limit does not exist, the improper integral of e^(-x)cos(x) dx from 0 to infinity diverges.