Let f ( x ) = e-x – 5. Does
converge or diverge?
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
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
either both converge or both diverge. Multiplying a function by a constant doesn't affect whether the integral of that function converges or diverges.