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

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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:

\(\ds \int \frac {\d x} {\sqrt {2 x - x^2} }\) \(=\) \(\ds \dfrac {-1} {\sqrt 1} \map \arcsin {\dfrac {2 \paren {-1} x + 2} {\sqrt {\size {2^2 - 4 \times \paren {-1} \times 0} } } } + C\) as $b^2 - 4 a c = 4 \ne 0$
\(\ds \) \(=\) \(\ds \dfrac {-1} {\sqrt 1} \map \arcsin {\dfrac {-2 x + 2} {\sqrt 4} } + C\) simplifying
\(\ds \) \(=\) \(\ds -\map \arcsin {1 - x} + C\) simplifying
\(\ds \) \(=\) \(\ds \map \arcsin {x - 1} + C\) Arcsine is Odd Function

$\blacksquare$


Sources

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