This antiderivative has arguable the highest ratio of “really hard to compute” to “really easy to write”:

As we’ve seen in this series, the answer is

Also, as long as and , there is an alternative answer:

.

In this concluding post of this series, I’d like to talk about the practical implications of the assumptions that and .

For the sake of simplicity for the rest of this post, let

and

.

If I evaluate a definite integral of over an interval that contains neither or , then either or can be used. Courtesy of Mathematica:

However, if the region of integration contains either or (or both), then only using returns the correct answer.

So this should be a cautionary tale about solving for angles, as the innocent-looking that appeared several posts ago ultimately makes a big difference in the final answers that are obtained.

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