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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