It looks like this integral probably converges, since *f* ( *x* ) looks a lot like *e*^{-x}, and we know converges. To show this for sure, we know that *e*^{-x} – 5 < *e*^{-x}.
Since *f* ( *x* ) < *e*^{-x} on [1,∞) and converges, converges also. There's one more property we should toss in. Let *c* be a constant. Then we know with definite integrals This extends to improper integrals where the function is badly behaved. It also extends to improper integrals where the limits are badly behaved. The integrals and either both converge or both diverge. Multiplying a function by a constant doesn't affect whether the integral of that function converges or diverges. |