Test the following improper integral for convergence:

The integral converges.

*Proof.* We recall the definition of the hyperbolic cosine in terms of the exponential,

Then we have

The middle integral converges since it is a proper integral of an integrable function. For the integral on the left we have

But, we have

for all . We know converges by Example 4 (page 418 of Apostol); hence,

converges. Since this is also the third integral in the sum above, we have proved that every integral in the sum converges, and hence we have established the convergence of

The integral should zero to infinity for the proof to have some value

very nice

When you made the substitution x=-t, you forgot a minus. Since the function x/coshx is odd, the result of integration is zero.