I need the following to complete a proof.
I've checked it in Maple and it's true. All I have left is to actually show it (without referring to Maple ofc).
Here is a third method using the Laplace transform
Consider the function
Is the integral you are looking for
Using a trig Identity we get
Now if we take the Laplace transform with respect to t we get
Now by partial fractions on the 2nd term we get
Plugging this back in gives
Now finally take the inverse transform to get