Primitive of Reciprocal of Root of a x squared plus b x plus c/Examples/2 + 4 x - 3 x^2
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Example of Use of Primitive of $\dfrac 1 {\sqrt {a x^2 + b x + c} }$
- $\ds \int \dfrac {\d x} {\sqrt {2 + 4 x - 3 x^2} } = \dfrac 1 {\sqrt 3} \map \arcsin {\dfrac {3 x - 2} {10} } + C$
Proof 1
| \(\ds \int \dfrac {\d x} {\sqrt {2 + 4 x - 3 x^2} }\) | \(=\) | \(\ds \dfrac {-1} {\sqrt 3} \map \arcsin {\dfrac {2 \times \paren {-3} x + 4} {\sqrt {\size {4^2 - 4 \times \paren {-3} \times 2} } } } + C\) | Primitive of $\dfrac 1 {\sqrt {a x^2 + b x + c} }$: $a < 0$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {-1} {\sqrt 3} \map \arcsin {\dfrac {-6 x + 4} {\sqrt {16 + 24} } } + C\) | simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 {\sqrt 3} \map \arcsin {\dfrac {2 \paren {3 x - 2} } {2 \sqrt {10} } } + C\) | Sine Function is Odd | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 {\sqrt 3} \map \arcsin {\dfrac {3 x - 2} {\sqrt {10} } } + C\) | simplifying |
$\blacksquare$
Proof 2
| \(\ds \int \dfrac {\d x} {\sqrt {2 + 4 x - 3 x^2} }\) | \(=\) | \(\ds \dfrac 1 {\sqrt 3} \int \dfrac {\d x} {\sqrt {\frac 2 3 + \frac 4 3 x - x^2} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 {\sqrt 3} \int \dfrac {\d x} {\sqrt {\frac {10} 9 - \paren {x - \frac 2 3}^2} }\) | Completing the Square | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 {\sqrt 3} \map \arcsin {\dfrac {x - \frac 2 3} {\sqrt {\frac {10} 9} } } + C\) | Primitive of $\dfrac 1 {\sqrt {a^2 - x^2} }$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 {\sqrt 3} \map \arcsin {\dfrac {3 x - 2} {\sqrt {10} } } + C\) | simplifying |
$\blacksquare$