Problem: Decide whether the following integral converges:
Solution: We know that there is no antiderivative for this function that we could write using elementary functions. Therefore we cannot evaluate this integral using standard methods. (By the way, one can find the value of this integral using a certain trick, it is half of the square root of π). We will try to use the Comparison test first, because it is easier (if it works) than the limit alternative.
What really bothers us in this integral is the square, without it it would be
a simple integral. So we try to get rid of it. Clearly, for
Since the test integral converges (see for instance here), by the Comparison test, also the smaller integral converges.
On the interval
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