This antiderivative has arguable the highest ratio of “really hard to compute” to “really easy to write”:
So far, I’ve shown that the denominator can be factored over the real numbers:
after finding the partial fractions decomposition.
Let me start with the first of the two integrals. It’d be nice to use the substitution . However,
, and so this substitution can’t be used cleanly. So, let me force the numerator to have this form, at least in part:
The substitution can now be applied to the first integral:
.
On the last line, I was able to remove the absolute value signs because is an irreducible quadratic and hence is never equal to zero for any real number
.
Similarly, I’ll try to apply the substitution to the second integral:
The substitution can now be applied to the first integral:
.
So, thus far, I have shown that
I’ll consider the evaluation of the remaining two integrals in tomorrow’s post.
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