Category:Primitive of Reciprocal of Root of a x squared plus b x plus c
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This category contains pages concerning Primitive of $\dfrac 1 {\sqrt {a x^2 + b x + c} }$:
$a$ greater than $0$
Let $a \in \R_{> 0}$.
Then for $x \in \R$ such that $a x^2 + b x + c > 0$:
- $\ds \int \frac {\d x} {\sqrt {a x^2 + b x + c} } = \begin {cases}
\dfrac 1 {\sqrt a} \ln \size {2 \sqrt a \sqrt {a x^2 + b x + c} + 2 a x + b} + C & : b^2 - 4 a c > 0 \\ \dfrac 1 {\sqrt a} \map \arsinh {\dfrac {2 a x + b} {\sqrt {4 a c - b^2} } } + C & : b^2 - 4 a c < 0 \\ \dfrac 1 {\sqrt a} \ln \size {2 a x + b} + C & : b^2 - 4 a c = 0 \end {cases}$ where $\arsinh$ denotes the area hyperbolic sine function.
$a$ less than $0$
Let $a \in \R_{< 0}$.
Then for $x \in \R$ such that $a x^2 + b x + c > 0$:
- $\ds \int \frac {\d x} {\sqrt {a x^2 + b x + c} } = \dfrac {-1} {\sqrt {-a} } \map \arcsin {\dfrac {2 a x + b} {\sqrt {\size {b^2 - 4 a c} } } } + C$
given that $b^2 \ne 4 a c$.
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Pages in category "Primitive of Reciprocal of Root of a x squared plus b x plus c"
The following 13 pages are in this category, out of 13 total.
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- Primitive of Reciprocal of Root of a x squared plus b x plus c
- Primitive of Reciprocal of Root of a x squared plus b x plus c/a equal to 0
- Primitive of Reciprocal of Root of a x squared plus b x plus c/a greater than 0
- Primitive of Reciprocal of Root of a x squared plus b x plus c/a greater than 0/Completing the Square
- Primitive of Reciprocal of Root of a x squared plus b x plus c/a greater than 0/Negative Discriminant
- Primitive of Reciprocal of Root of a x squared plus b x plus c/a greater than 0/Positive Discriminant
- Primitive of Reciprocal of Root of a x squared plus b x plus c/a greater than 0/Zero Discriminant
- Primitive of Reciprocal of Root of a x squared plus b x plus c/a less than 0
- Primitive of Reciprocal of Root of a x squared plus b x plus c/a less than 0/Completing the Square
- Primitive of Reciprocal of Root of a x squared plus b x plus c/a less than 0/Negative Discriminant
- Primitive of Reciprocal of Root of a x squared plus b x plus c/a less than 0/Positive Discriminant
- Primitive of Reciprocal of Root of a x squared plus b x plus c/a less than 0/Zero Discriminant
- Primitive of Reciprocal of Root of a x squared plus b x plus c/Examples/2 x + x^2