Previously in this series, I showed that
My wife had asked me to compute this integral by hand because Mathematica 4 and Mathematica 8 gave different answers. At the time, I eventually obtained the solution by multiplying the top and bottom of the integrand by
and then employing the substitution
(after using trig identities to adjust the limits of integration).
But this wasn’t the only method I tried. Indeed, I tried two or three different methods before deciding they were too messy and trying something different. So, for the rest of this series, I’d like to explore different ways that the above integral can be computed.
So far, I have shown that
To simplify the denominator even further, I will combine the two trigonometric terms in the denominator; this is possible because the argument of both the sine and cosine functions are the same. To this end, notice that
Next, let be the unique angle so that
With this substitution, we find that
Therefore, the integral may be rewritten as
where and .
I’ll continue this different method of evaluating this integral in tomorrow’s post.