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.

### Like this:

Like Loading...

*Related*

## 1 Comment