Primitive of Reciprocal of Root of a x squared plus b x plus c/Examples/2 x - x^2

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

 * $\ds \int \dfrac {\d x} {\sqrt {2 x - x^2} } = \map \arcsin {x - 1} + C$

Proof
From Primitive of $\dfrac 1 {\sqrt {a x^2 + b x + c} }$ for negative $a$:
 * $\ds \int \frac {\d x} {\sqrt {a x^2 + b x + c} } = \dfrac {-1} {\sqrt {-a} } \map \arcsin {\dfrac {2 a x + b} {\sqrt {\size {b^2 - 4 a c} } } } + C$

as long as $b^2 - 4 a c \ne 0$.

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