Primitive of Reciprocal of a x squared plus b x plus c/Examples/x^2 + 4 x + 5

Example of Use of Primitive of $\dfrac 1 {a x^2 + b x + c}$

 * $\ds \int \dfrac {\d x} {x^2 + 4 x + 5} = \map \arctan {x + 2} + C$

Proof
We aim to use Primitive of $\dfrac 1 {a x^2 + b x + c}$ with:

We note that:

Hence from Primitive of $\dfrac 1 {a x^2 + b x + c}$:
 * $\ds \int \frac {\d x} {a x^2 + b x + c} = \dfrac 2 {\sqrt {4 a c - b^2} } \map \arctan {\dfrac {2 a x + b} {\sqrt {4 a c - b^2} } } + C$

Substituting for $a$, $b$ and $c$ and simplifying: