Primitive of Reciprocal of x by Root of a x squared plus b x plus c/Proof 1

Lemma
Let $x > 0$, and so $u > 0$.

Then we have:


 * $\ds \int \frac {\d x} {x \sqrt {a x^2 + b x + c} } = -\int \frac {\d u} {\sqrt {a + b u + c u^2} }$

We consider the two cases where $c > 0$ and $c < 0$.

First we take $c > 0$.

Thus:

Then we take $c < 0$.

Thus:

Now we consider what happens when $x < 0$, and so $u < 0$.

We have:

which leads us to:

Finally we take $c < 0$.

Thus: