hcju/proofwikips huggingface dataset:
Primitive of x squared over x squared minus a squared squared
:$\displaystyle \int \frac {x^2 \ \mathrm d x} {\left({x^2 - a^2}\right)^2} = \frac {-x} {2 \left({x^2 - a^2}\right)} + \frac 1 {4 a} \ln \left({\frac {x - a} {x + a} }\right) + C$ for $x^2 > a^2$.
Primitive of x cubed over x squared minus a squared squared
:$\displaystyle \int \frac {x^3 \ \mathrm d x} {\left({x^2 - a^2}\right)^2} = \frac {-a^2} {2 \left({x^2 - a^2}\right)} + \frac 1 2 \ln \left({x^2 - a^2}\right) + C$ for $x^2 > a^2$.
Primitive of Reciprocal of x by x squared minus a squared squared
:$\displaystyle \int \frac {\d x} {x \paren {x^2 - a^2}^2} = \frac {-1} {2 a^2 \left({x^2 - a^2}\right)} + \frac 1 {2 a^4} \ln \left({\frac {x^2} {x^2 - a^2} }\right) + C$ for $x^2 > a^2$.
Primitive of Reciprocal of x squared by x squared minus a squared squared
:$\displaystyle \int \frac {\mathrm d x} {x^2 \left({x^2 - a^2}\right)^2} = \frac {-1} {a^4 x} - \frac x {2 a^4 \left({x^2 - a^2}\right)} + \frac 3 {4 a^5} \ln \left({\frac {x + a} {x - a} }\right) + C$ for $x^2 > a^2$.
Primitive of Reciprocal of x cubed by x squared minus a squared squared
:$\ds \int \frac {\d x} {x^3 \paren {x^2 - a^2}^2} = \frac {-1} {2 a^4 x^2} - \frac 1 {2 a^4 \paren {x^2 - a^2} } + \frac 1 {a^6} \map \ln {\frac {x^2} {x^2 - a^2} } + C$ for $x^2 > a^2$.
Primitive of Reciprocal of Power of x squared minus a squared
:$\displaystyle \int \frac {\mathrm d x} {\left({x^2 - a^2}\right)^n} = \frac {-x} {2 \left({n - 1}\right) a^2 \left({x^2 - a^2}\right)^{n - 1} } - \frac {2 n - 3} {\left({2 n - 2}\right) a^2} \int \frac {\mathrm d x} {\left({x^2 - a^2}\right)^{n - 1} }$ for $x^2 > a^2$.
Primitive of x over Power of x squared minus a squared
:$\displaystyle \int \frac {x \rd x} {\paren {x^2 - a^2}^n} = \frac {-1} {2 \paren {n - 1} \paren {x^2 - a^2}^{n - 1} }$ for $x^2 > a^2$.
Primitive of Power of x over Power of x squared minus a squared
:$\displaystyle \int \frac {x^m \rd x} {\paren {x^2 - a^2}^n} = \int \frac {x^{m - 2} \rd x} {\paren {x^2 - a^2}^{n - 1} } + a^2 \int \frac {x^{m - 2} \rd x} {\paren {x^2 - a^2}^n}$ for $x^2 > a^2$.
Sign of Quotient of Factors of Difference of Squares
Let $a, b \in \R$ such that $a \ne b$. Then :$\map \sgn {a^2 - b^2} = \map \sgn {\dfrac {a + b} {a - b} } = \map \sgn {\dfrac {a - b} {a + b} }$ where $\sgn$ denotes the signum of a real number.
Signum Function is Completely Multiplicative
The signum function on the set of real numbers is a completely multiplicative function: :$\forall x, y \in \R: \map \sgn {x y} = \map \sgn x \map \sgn y$
Absolute Value Function is Completely Multiplicative
The absolute value function on the real numbers $\R$ is completely multiplicative: :$\forall x, y \in \R: \left\vert{x y}\right\vert = \left\vert{x}\right\vert \, \left\vert{y}\right\vert$ where $\left \vert{a}\right \vert$ denotes the absolute value of $a$.
Signum Function of Reciprocal
Let $x \in \R$ such that $x \ne 0$. Then: :$\map \sgn x = \map \sgn {\dfrac 1 x}$ where $\map \sgn x$ denotes the signum of $x$.
Primitive of x over a squared minus x squared
:$\displaystyle \int \frac {x \rd x} {a^2 - x^2} = -\frac 1 2 \, \map \ln {a^2 - x^2} + C$ for $x^2 < a^2$.
Linear Combination of Laplace Transforms
Then: :$\laptrans {\lambda \, \map f t + \mu \, \map g t} = \lambda \laptrans {\map f t} + \mu \laptrans {\map g t}$ everywhere all the above expressions are defined.
Primitive of x cubed over a squared minus x squared
:$\displaystyle \int \frac {x^3 \ \mathrm d x} {a^2 - x^2} = -\frac {x^2} 2 - \frac {a^2} 2 \ln \left({a^2 - x^2}\right) + C$ for $x^2 < a^2$.
Primitive of Reciprocal of x squared by a squared minus x squared
:$\displaystyle \int \frac {\d x} {x^2 \paren {a^2 - x^2} } = \frac {-1} {a^2 x} + \frac 1 {2 a^3} \map \ln {\frac {a + x} {a - x} } + C$ for $x^2 < a^2$.
Primitive of x over a squared minus x squared squared
:$\displaystyle \int \frac {x \ \mathrm d x} {\left({a^2 - x^2}\right)^2} = \frac 1 {2 \left({a^2 - x^2}\right)} + C$ for $x^2 < a^2$.
Primitive of x squared over a squared minus x squared squared
:$\displaystyle \int \frac {x^2 \rd x} {\paren {a^2 - x^2}^2} = \frac x {2 \paren {a^2 - x^2} } - \frac 1 {4 a} \map \ln {\frac {a + x} {a - x} } + C$ for $x^2 < a^2$.
Primitive of x cubed over a squared minus x squared squared
:$\displaystyle \int \frac {x^3 \rd x} {\paren {a^2 - x^2}^2} = \frac {a^2} {2 \paren {a^2 - x^2} } + \frac 1 2 \map \ln {a^2 - x^2} + C$ for $x^2 < a^2$.
Primitive of Reciprocal of x by a squared minus x squared squared
:$\displaystyle \int \frac {\d x} {x \paren {a^2 - x^2}^2} = \frac 1 {2 a^2 \paren {a^2 - x^2} } + \frac 1 {2 a^4} \map \ln {\frac {x^2} {a^2 - x^2} } + C$ for $x^2 < a^2$.
Primitive of Reciprocal of x squared by a squared minus x squared squared
:$\displaystyle \int \frac {\d x} {x^2 \paren {a^2 - x^2}^2} = \frac {-1} {a^4 x} + \frac x {2 a^4 \paren {a^2 - x^2} } + \frac 3 {4 a^5} \map \ln {\frac {a + x} {a - x} } + C$ for $x^2 < a^2$.
Primitive of Reciprocal of Power of a squared minus x squared
:$\displaystyle \int \frac {\mathrm d x} {\left({a^2 - x^2}\right)^n} = \frac x {2 \left({n - 1}\right) a^2 \left({a^2 - x^2}\right)^{n - 1} } + \frac {2 n - 3} {\left({2 n - 2}\right) a^2} \int \frac {\mathrm d x} {\left({a^2 - x^2}\right)^{n - 1} }$ for $x^2 > a^2$.
Primitive of Power of x over Power of a squared minus x squared
:$\displaystyle \int \frac {x^m \rd x} {\paren {a^2 - x^2}^n} = a^2 \int \frac {x^{m - 2} \rd x} {\paren {a^2 - x^2}^n} - \int \frac {x^{m - 2} \rd x} {\paren {a^2 - x^2}^{n - 1} }$ for $x^2 < a^2$.
Primitive of Reciprocal of Power of x by Power of a squared minus x squared
:$\displaystyle \int \frac {\mathrm d x} {x^m \left({a^2 - x^2}\right)^n} = \frac 1 {a^2} \int \frac {\mathrm d x} {x^m \left({a^2 - x^2}\right)^{n-1} } + \frac 1 {a^2} \int \frac {\mathrm d x} {x^{m-2} \left({a^2 - x^2}\right)^n}$ for $x^2 < a^2$.
Primitive of x over Root of x squared plus a squared
:$\ds \int \frac {x \rd x} {\sqrt {x^2 + a^2} } = \sqrt {x^2 + a^2} + C$
Primitive of x cubed over Root of x squared plus a squared
:$\displaystyle \int \frac {x^2 \rd x} {\sqrt {x^2 + a^2} } = \frac {\paren {\sqrt {x^2 + a^2} }^3} 3 - a^2 \sqrt {x^2 + a^2} + C$
Primitive of Reciprocal of x squared by Root of x squared plus a squared
:$\displaystyle \int \frac {\mathrm d x} {x^2 \sqrt {x^2 + a^2} } = \frac {-\sqrt {x^2 + a^2} } {a^2 x} + C$
Primitive of Reciprocal of x cubed by Root of x squared plus a squared
:$\displaystyle \int \frac {\mathrm d x} {x^3 \sqrt {x^2 + a^2} } = \frac {- \sqrt {x^2 + a^2} } {2 a^2 x^2} + \frac 1 {2 a^3} \ln \left({\frac {a + \sqrt {x^2 + a^2} } x}\right) + C$
Primitive of x by Root of x squared plus a squared
:$\displaystyle \int x \sqrt {x^2 + a^2} \rd x = \frac {\paren {\sqrt {x^2 + a^2} }^3} 3 + C$
Primitive of Root of x squared plus a squared over x
:$\displaystyle \int \frac {\sqrt {x^2 + a^2} } x \ \mathrm d x = \sqrt {x^2 + a^2} - a \ln \left({\frac {a + \sqrt {x^2 + a^2} } a}\right) + C$
Primitive of Root of x squared plus a squared over x squared
:$\displaystyle \int \frac {\sqrt {x^2 + a^2} } {x^2} \rd x = \frac {-\sqrt {x^2 + a^2} } x + \map \ln {x + \sqrt {x^2 + a^2} } + C$
Primitive of Root of x squared plus a squared over x cubed
:$\displaystyle \int \frac {\sqrt {x^2 + a^2} } {x^3} \ \mathrm d x = \frac {-\sqrt {x^2 + a^2} } {2 x^2} - \frac 1 {2 a} \ln \left({\frac {a + \sqrt {x^2 + a^2} } x}\right) + C$
Primitive of Reciprocal of Root of x squared plus a squared cubed
:$\ds \int \frac {\d x} {\paren {\sqrt {x^2 + a^2} }^3} = \frac x {a^2 \sqrt {x^2 + a^2} } + C$
Primitive of x over Root of x squared plus a squared cubed
:$\displaystyle \int \frac {x \rd x} {\paren {\sqrt {x^2 + a^2} }^3} = \frac {-1} {\sqrt {x^2 + a^2} } + C$
Primitive of Reciprocal of x by Root of x squared plus a squared cubed
:$\displaystyle \int \frac {\d x} {x \paren {\sqrt {x^2 + a^2} }^3} = \frac 1 {a^2 \sqrt {x^2 + a^2} } - \frac 1 {a^3} \map \ln {\frac {a + \sqrt {x^2 + a^2} } x} + C$
Primitive of Reciprocal of x cubed by Root of x squared plus a squared cubed
:$\displaystyle \int \frac {\mathrm d x} {x^3 \left({\sqrt {x^2 + a^2} }\right)^3} = \frac {-1} {2 a^2 x^2 \sqrt {x^2 + a^2} } - \frac 3 {2 a^4 \sqrt {x^2 + a^2} } + \frac 3 {2 a^5} \ln \left({\frac {a + \sqrt {x^2 + a^2} } x}\right) + C$
Primitive of Root of x squared plus a squared cubed
:$\ds \int \paren {\sqrt {x^2 + a^2} }^3 \rd x = \frac {x \paren {\sqrt {x^2 + a^2} }^3} 4 + \frac {3 a^2 x \sqrt {x^2 + a^2} } 8 + \frac {3 a^4} 8 \map \ln {x + \sqrt {x^2 + a^2} } + C$
Primitive of x by Root of x squared plus a squared cubed
:$\displaystyle \int x \left({\sqrt {x^2 + a^2} }\right)^3 \ \mathrm d x = \frac {\left({\sqrt {x^2 + a^2} }\right)^5} 5 + C$
Primitive of x squared by Root of x squared plus a squared cubed
:$\displaystyle \int x^2 \left({\sqrt {x^2 + a^2} }\right)^3 \ \mathrm d x = \frac {x \left({\sqrt {x^2 + a^2} }\right)^5} 6 - \frac {a^2 x \left({\sqrt {x^2 + a^2} }\right)^3} {24} - \frac {a^4 x \sqrt {x^2 + a^2} } {16} - \frac {a^6} {16} \ln \left({x + \sqrt {x^2 + a^2} }\right) + C$
Primitive of x cubed by Root of x squared plus a squared cubed
:$\displaystyle \int x^3 \paren {\sqrt {x^2 + a^2} }^3 \rd x = \frac {\paren {\sqrt {x^2 + a^2} }^7} 7 - \frac {a^2 \paren {\sqrt {x^2 + a^2} }^5} 5 + C$
Primitive of x over Root of x squared minus a squared
:$\displaystyle \int \frac {x \rd x} {\sqrt {x^2 - a^2} } = \sqrt {x^2 - a^2} + C$
Primitive of x squared over Root of x squared minus a squared
{{:Primitive of x squared over Root of x squared minus a squared/Inverse Hyperbolic Cosine Form}}
Primitive of x cubed over Root of x squared minus a squared
:$\displaystyle \int \frac {x^3 \rd x} {\sqrt {x^2 - a^2} } = \frac {\paren {\sqrt {x^2 - a^2} }^3} 3 + a^2 \sqrt {x^2 - a^2} + C$
Primitive of Reciprocal of x squared by Root of x squared minus a squared
:$\displaystyle \int \frac {\d x} {x^2 \sqrt {x^2 - a^2} } = \frac {\sqrt {x^2 - a^2} } {a^2 x} + C$
Primitive of Reciprocal of x cubed by Root of x squared minus a squared
:$\displaystyle \int \frac {\d x} {x^3 \sqrt {x^2 - a^2} } = \frac {\sqrt {x^2 - a^2} } {2 a^2 x^2} + \frac 1 {2 a^3} \arcsec \size {\frac x a} + C$
Primitive of Root of x squared minus a squared
{{:Primitive of Root of x squared minus a squared/Inverse Hyperbolic Cosine Form}}
Primitive of x by Root of x squared minus a squared
:$\displaystyle \int x \sqrt {x^2 - a^2} \rd x = \frac {\paren {\sqrt {x^2 - a^2} }^3} 3 + C$
Primitive of x squared by Root of x squared minus a squared
:$\displaystyle \int x^2 \sqrt {x^2 - a^2} \rd x = \frac {x \paren {\sqrt {x^2 - a^2} }^3} 4 + \frac {a^2 x \sqrt {x^2 - a^2} } 8 - \frac {a^4} 8 \ln \size {x + \sqrt {x^2 - a^2} } + C$
Primitive of Root of x squared minus a squared over x
:$\displaystyle \int \frac {\sqrt {x^2 - a^2} } x \rd x = \sqrt {x^2 - a^2} - \frac 1 {2 a} \arcsec \size {\frac x a} + C$
Primitive of Root of x squared minus a squared over x cubed
:$\displaystyle \int \frac {\sqrt {x^2 - a^2} } {x^3} \rd x = \frac {-\sqrt {x^2 - a^2} } {2 x^2} + \frac 1 {2 a} \arcsec \size {\frac x a} + C$
Primitive of Reciprocal of Root of x squared minus a squared cubed
:$\displaystyle \int \frac {\d x} {\paren {\sqrt {x^2 - a^2} }^3} = \frac {-x} {a^2 \sqrt {x^2 - a^2} } + C$
Primitive of x over Root of x squared minus a squared cubed
:$\displaystyle \int \frac {x \rd x} {\paren {\sqrt {x^2 - a^2} }^3} = \frac {-1} {\sqrt {x^2 - a^2} } + C$
Primitive of x squared over Root of x squared minus a squared cubed
:$\displaystyle \int \frac {x^2 \rd x} {\paren {\sqrt {x^2 - a^2} }^3} = \frac {-x} {\sqrt {x^2 - a^2} } + \ln \size {x + \sqrt {x^2 - a^2} } + C$
Primitive of x cubed over Root of x squared minus a squared cubed
:$\displaystyle \int \frac {x^3 \rd x} {\paren {\sqrt {x^2 - a^2} }^3} = \sqrt {x^2 - a^2} - \frac {a^2} {\sqrt {x^2 - a^2} } + C$
Primitive of Reciprocal of x by Root of x squared minus a squared cubed
:$\displaystyle \int \frac {\d x} {x \paren {\sqrt {x^2 - a^2} }^3} = \frac {-1} {a^2 \sqrt {x^2 - a^2} } - \frac 1 {a^3} \arcsec \size {\frac x a} + C$
Primitive of Reciprocal of x cubed by Root of x squared minus a squared cubed
:$\displaystyle \int \frac {\d x} {x^3 \paren {\sqrt {x^2 - a^2} }^3} = \frac 1 {2 a^2 x^2 \sqrt {x^2 - a^2} } - \frac 3 {2 a^4 \sqrt {x^2 - a^2} } + \frac 3 {2 a^5} \arcsec \size {\frac x a} + C$
Primitive of Root of x squared minus a squared cubed
:$\displaystyle \int \paren {\sqrt {x^2 - a^2} }^3 \rd x = \frac {x \paren {\sqrt {x^2 - a^2} }^3} 4 - \frac {3 a^2 x \sqrt {x^2 - a^2} } 8 + \frac {3 a^4} 8 \ln \size {x + \sqrt {x^2 - a^2} } + C$
Primitive of x by Root of x squared minus a squared cubed
:$\displaystyle \int x \paren {\sqrt {x^2 - a^2} }^3 \rd x = \frac {\paren {\sqrt {x^2 - a^2} }^5} 5 + C$
Primitive of x cubed by Root of x squared minus a squared cubed
:$\displaystyle \int x^3 \paren {\sqrt {x^2 - a^2} }^3 \rd x = \frac {\paren {\sqrt {x^2 - a^2} }^7} 7 + \frac {a^2 \paren {\sqrt {x^2 - a^2} }^5} 5 + C$
Primitive of Root of x squared minus a squared cubed over x squared
:$\displaystyle \int \frac {\paren {\sqrt {x^2 - a^2} }^3} {x^2} \rd x = \frac {-\paren {\sqrt {x^2 - a^2} }^3} x + \frac{3 x \sqrt {x^2 - a^2} } 2 - \frac {3 a^2} 2 \ln \size {x + \sqrt {x^2 - a^2} } + C$
Primitive of Root of x squared minus a squared cubed over x cubed
:$\displaystyle \int \frac {\paren {\sqrt {x^2 - a^2} }^3} {x^3} \rd x = \frac {-\paren {\sqrt {x^2 - a^2} }^3} {2 x^2} + \frac {3 \sqrt {x^2 - a^2} } 2 - \frac {3 a} 2 \arcsec \size {\frac x a} + C$
Primitive of x over Root of a squared minus x squared
:$\displaystyle \int \frac {x \rd x} {\sqrt {a^2 - x^2} } = -\sqrt {a^2 - x^2} + C$
Primitive of x cubed over Root of a squared minus x squared
:$\displaystyle \int \frac {x^3 \rd x} {\sqrt {a^2 - x^2} } = \frac {\left({\sqrt {a^2 - x^2} }\right)^3} 3 - a^2 \sqrt {a^2 - x^2} + C$
Primitive of Reciprocal of x squared by Root of a squared minus x squared
:$\displaystyle \int \frac {\mathrm d x} {x^2 \sqrt {a^2 - x^2} } = \frac {-\sqrt {a^2 - x^2} } {a^2 x} + C$
Primitive of Reciprocal of x cubed by Root of a squared minus x squared
:$\displaystyle \int \frac {\d x} {x^3 \sqrt {a^2 - x^2} } = \frac {-\sqrt {a^2 - x^2} } {2 a^2 x^2} - \frac 1 {2 a^3} \map \ln {\frac {a + \sqrt {a^2 - x^2} } x} + C$
Primitive of Root of a squared minus x squared
:$\displaystyle \int \sqrt {a^2 - x^2} \rd x = \frac {x \sqrt {a^2 - x^2} } 2 + \frac {a^2} 2 \arcsin \frac x a + C$
Primitive of x by Root of a squared minus x squared
:$\displaystyle \int x \sqrt {a^2 - x^2} \ \mathrm d x = \frac {-\left({\sqrt {a^2 - x^2} }\right)^3} 3 + C$
Primitive of x cubed by Root of a squared minus x squared
:$\displaystyle \int x^3 \sqrt {a^2 - x^2} \ \mathrm d x = \frac {\left({\sqrt {a^2 - x^2} }\right)^5} 5 - \frac {a^2 \left({\sqrt {a^2 - x^2} }\right)^3} 3 + C$
Primitive of Root of a squared minus x squared over x
:$\displaystyle \int \frac {\sqrt {a^2 - x^2} } x \rd x = \sqrt {a^2 - x^2} - a \, \map \ln {\frac {a + \sqrt {a^2 - x^2} } x} + C$
Primitive of Root of a squared minus x squared over x squared
:$\displaystyle \int \frac {\sqrt {a^2 - x^2} } {x^2} \rd x = \frac {-\sqrt {a^2 - x^2} } x - \arcsin \frac x a + C$
Primitive of Reciprocal of Root of a squared minus x squared cubed
:$\displaystyle \int \frac {\mathrm d x} {\left({\sqrt {a^2 - x^2} }\right)^3} = \frac x {a^2 \sqrt {a^2 - x^2} } + C$
Primitive of x over Root of a squared minus x squared cubed
:$\displaystyle \int \frac {x \ \mathrm d x} {\left({\sqrt {a^2 - x^2} }\right)^3} = \frac 1 {\sqrt {a^2 - x^2} } + C$
Primitive of x squared over Root of a squared minus x squared cubed
:$\displaystyle \int \frac {x^2 \rd x} {\paren {\sqrt {a^2 - x^2} }^3} = \frac x {\sqrt {a^2 - x^2} } - \arcsin \frac x a + C$
Primitive of Reciprocal of x by Root of a squared minus x squared cubed
:$\displaystyle \int \frac {\mathrm d x} {x \left({\sqrt {a^2 - x^2} }\right)^3} = \frac 1 {a^2 \sqrt {a^2 - x^2} } - \frac 1 {a^3} \ln \left({\frac {a + \sqrt {a^2 - x^2} } x}\right) + C$
Primitive of Reciprocal of x squared by Root of a squared minus x squared cubed
:$\displaystyle \int \frac {\mathrm d x} {x^2 \left({\sqrt {a^2 - x^2} }\right)^3} = \frac {-\sqrt {a^2 - x^2} } {a^4 x} + \frac x {a^4 \sqrt {a^2 - x^2} } + C$
Primitive of Root of a squared minus x squared cubed
:$\displaystyle \int \paren {\sqrt {a^2 - x^2} }^3 \rd x = \frac {x \paren {\sqrt {a^2 - x^2} }^3} 4 + \frac {3 a^2 x \sqrt {a^2 - x^2} } 8 + \frac {3 a^4} 8 \arcsin \frac x a + C$
Primitive of x by Root of a squared minus x squared cubed
:$\displaystyle \int x \left({\sqrt {a^2 - x^2} }\right)^3 \ \mathrm d x = \frac {-\left({\sqrt {a^2 - x^2} }\right)^5} 5 + C$
Primitive of x squared by Root of a squared minus x squared cubed
:$\displaystyle \int x^2 \paren {\sqrt {a^2 - x^2} }^3 \rd x = \frac {-x \paren {\sqrt {a^2 - x^2} }^5} 6 + \frac {a^2 x \paren {\sqrt {a^2 - x^2} }^3} {24} + \frac {a^4 x \sqrt {a^2 - x^2} } {16} + \frac {a^6} {16} \arcsin \frac x a + C$
Primitive of Reciprocal of a x squared plus b x plus c
:$\displaystyle \int \frac {\d x} {a x^2 + b x + c} = \begin{cases} \dfrac 2 {\sqrt {4 a c - b^2} } \, \map \arctan {\dfrac {2 a x + b} {\sqrt {4 a c - b^2} } } + C & : b^2 - 4 a c < 0 \\ \dfrac 1 {\sqrt {b^2 - 4 a c} } \ln \size {\dfrac {2 a x + b - \sqrt {b^2 - 4 a c} } {2 a x + b + \sqrt {b^2 - 4 a c} } } + C & : b^2 - 4 a c > 0 \\ \dfrac {-2} {2 a x + b} + C & : b^2 = 4 a c \end{cases}$
Primitive of Reciprocal of a x squared plus b x plus c/a equal to 0
:$\displaystyle \int \frac {\d x} {a x^2 + b x + c} = \frac 1 b \ln \size {b x + c} + C$ when $a = 0$.
Primitive of Reciprocal of a x squared plus b x plus c/b equal to 0/Proof 2
Let $a \in \R_{\ne 0}$. Let $b = 0$. Then: :$\displaystyle \int \frac {\mathrm d x} {a x^2 + b x + c} = \begin{cases} \dfrac 1 {\sqrt {a c} } \arctan \left({x \sqrt {\dfrac a c} }\right) + C & : a c > 0 \\ \dfrac 1 {2 \sqrt {-a c} } \ln \left\vert{\dfrac {a x - \sqrt {-a c} } {a x + \sqrt {-a c} } }\right\vert + C & : a c < 0 \\ \dfrac {-1} {a x} + C & : c = 0 \end{cases}$
Primitive of x over a x squared plus b x plus c
Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {x \ \mathrm d x} {a x^2 + b x + c} = \frac 1 {2 a} \ln \left\vert{a x^2 + b x + c}\right\vert - \frac b {2 a} \int \frac {\mathrm d x} {a x^2 + b x + c}$
Primitive of x squared over a x squared plus b x plus c
Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {x^2 \rd x} {a x^2 + b x + c} = \frac x a - \frac b {2 a^2} \ln \size {a x^2 + b x + c} - \frac {b^2 - 2 a c} {2 a^2} \int \frac {\d x} {a x^2 + b x + c}$
Primitive of Reciprocal of a x squared plus b x plus c/Negative Discriminant
Let $a \in \R_{\ne 0}$. Let $b^2 - 4 a c < 0$. Then: :$\displaystyle \int \frac {\d x} {a x^2 + b x + c} = \frac 2 {\sqrt {4 a c - b^2} } \map \arctan {\dfrac {2 a x + b} {\sqrt {4 a c - b^2} } } + C$
Primitive of Reciprocal of a x squared plus b x plus c/Positive Discriminant
Let $a \in \R_{\ne 0}$. Let $b^2 - 4 a c > 0$. Then: :$\displaystyle \int \frac {\d x} {a x^2 + b x + c} = \frac 1 {\sqrt {b^2 - 4 a c} } \ln \size {\dfrac {2 a x + b - \sqrt {b^2 - 4 a c} } {2 a x + b + \sqrt {b^2 - 4 a c} } } + C$
Primitive of Reciprocal of a x squared plus b x plus c/Zero Discriminant
Let $a \in \R_{\ne 0}$. Let $b^2 - 4 a c = 0$. Then: :$\displaystyle \int \frac {\d x} {a x^2 + b x + c} = \frac {-2} {2 a x + b} + C$
Primitive of Reciprocal of square of a x squared plus b x plus c
Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {\mathrm d x} {\left({a x^2 + b x + c}\right)^2} = \frac {2 a x + b} {\left({4 a c - b^2}\right) \left({a x^2 + b x + c}\right)} + \frac {2 a} {4 a c - b^2} \int \frac {\mathrm d x} {a x^2 + b x + c}$
Completing the Square
Let $a, b, c, x$ be real numbers with $a \ne 0$. Then: :$a x^2 + b x + c = \dfrac {\paren {2 a x + b}^2 + 4 a c - b^2} {4 a}$ This process is known as '''completing the square'''.
Primitive of x over square of a x squared plus b x plus c
Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {x \ \mathrm d x} {\left({a x^2 + b x + c}\right)^2} = \frac {- \left({b x + 2 c}\right)} {\left({4 a c - b^2}\right) \left({a x^2 + b x + c}\right)} - \frac b {4 a c - b^2} \int \frac {\mathrm d x} {a x^2 + b x + c}$
Primitive of x squared over square of a x squared plus b x plus c
Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {x^2 \ \mathrm d x} {\left({a x^2 + b x + c}\right)^2} = \frac {\left({b^2 - 2 a c}\right) x + b c} {a \left({4 a c - b^2}\right) \left({a x^2 + b x + c}\right)} + \frac {2 c} {4 a c - b^2} \int \frac {\mathrm d x} {a x^2 + b x + c}$
Primitive of Power of x over Power of a x squared plus b x plus c
Let $a \in \R_{\ne 0}$. Then: {{begin-eqn}} {{eqn | l = \int \frac {x^m \rd x} {\paren {a x^2 + b x + c}^n}       | r = \frac {x^{m - 1} } {\paren {2 n - m - 1} a \paren {a x^2 + b x + c}^{n - 1} }       | c =  }} {{eqn | o =       | ro= +       | r = \frac {\paren {m - 1} c} {\paren {2 n - m - 1} a} \int \frac {x^{m - 2} \rd x} {\paren {a x^2 + b x + c}^n}       | c =  }} {{eqn | o =       | ro= -       | r = \frac {\paren {n - m} b} {\paren {2 n - m - 1} a} \int \frac {x^{m - 1} \rd x} {\paren {a x^2 + b x + c}^n}       | c =  }} {{end-eqn}}
Primitive of Reciprocal of x by square of a x squared plus b x plus c
Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {\d x} {x \paren {a x^2 + b x + c}^2} = \frac 1 {2 c \paren {a x^2 + b x + c} } - \frac b {2 c} \int \frac {\d x} {\paren {a x^2 + b x + c}^2} + \frac 1 c \int \frac {\d x} {x \paren {a x^2 + b x + c} }$
Primitive of Reciprocal of x squared by square of a x squared plus b x plus c
Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {\d x} {x^2 \paren {a x^2 + b x + c}^2} = \frac {-1} {c x \paren {a x^2 + b x + c} } - \frac {3 a} c \int \frac {\d x} {\paren {a x^2 + b x + c}^2} - \frac {2 b} c \int \frac {\d x} {x \paren {a x^2 + b x + c}^2}$
Primitive of Reciprocal of Power of x by Power of a x squared plus b x plus c
Let $a \in \R_{\ne 0}$. Then: {{begin-eqn}} {{eqn | l = \int \frac {\d x} {x^m \paren {a x^2 + b x + c}^n}       | r = \frac {-1} {\paren {m - 1} c x^{m - 1} \paren {a x^2 + b x + c}^{n - 1} } }} {{eqn | o =       | ro= -       | r = \frac {\paren {m + 2 n - 3} a} {\paren {m - 1} c} \int \frac {\d x} {x^{m - 2} \paren {a x^2 + b x + c}^n} }} {{eqn | o =       | ro= -       | r = \frac {\paren {m - n + 2} b} {\paren {m - 1} c} \int \frac {\d x} {x^{m - 1} \paren {a x^2 + b x + c}^n} }} {{end-eqn}}
Ring of Sets Closed under Finite Union
Let $\mathcal R$ be a ring of sets. Let $A_1, A_2, \ldots, A_n \in \mathcal R$. Then: :$\displaystyle \bigcup_{j \mathop = 1}^n A_j \in \mathcal R$
Sigma-Ring contains Limit Superior of Sequence of Sets
Let $\mathcal R$ be a $\sigma$-ring. Let $\left \langle{A_n}\right \rangle_{n \mathop \in \N} \in \mathcal R$ be a sequence of sets in $\mathcal R$. Then: :$\displaystyle \limsup_{n \to \infty} \ A_n \in \mathcal R$
Gamma Function for Non-Negative Integer Argument
The Gamma function satisfies: :$\map \Gamma z = \dfrac {\map \Gamma {z + 1} } z$ for any $z$ which is not a nonpositive integer.
Gamma Function of Positive Half-Integer
{{begin-eqn}} {{eqn | l = \Gamma \left({m + \frac 1 2}\right)       | r = \frac {\left({2 m}\right)!} {2^{2 m} m!} \sqrt \pi       | c =  }} {{eqn | r = \frac {1 \times 3 \times 5 \times \cdots \times \left({2 m - 1}\right)} {2^m} \sqrt \pi       | c =  }} {{end-eqn}} where: : $m + \dfrac 1 2$ is a half-integer such that $m > 0$ : $\Gamma$ denotes the Gamma function.
Gamma Function of Negative Half-Integer
{{begin-eqn}} {{eqn | l = \map \Gamma {-m + \frac 1 2}       | r = \frac {\paren {-1}^m 2^{2 m} m!} {\paren {2 m}!} \sqrt \pi       | c =  }} {{eqn | r = \frac {\paren {-1}^m 2^m} {1 \times 3 \times 5 \times \cdots \times \paren {2 m - 1} } \sqrt \pi       | c =  }} {{end-eqn}} where: :$-m + \dfrac 1 2$ is a half-integer such that $m > 0$ :$\Gamma$ denotes the Gamma function.
Reciprocal times Derivative of Gamma Function
Let $\Gamma$ denote the Gamma function. Then: :$\displaystyle \dfrac {\Gamma\,' \left({z}\right)} {\Gamma \left({z}\right)} = -\gamma + \sum_{n \mathop = 1}^\infty \left({\frac 1 n - \frac 1 {z + n - 1} }\right)$ where: : $\Gamma\,' \left({z}\right)$ denotes the derivative of the Gamma function : $\gamma$ denotes the Euler-Mascheroni constant.