hcju/proofwikips huggingface dataset:
Derivative of Gamma Function at 1
Let $\Gamma$ denote the Gamma function. Then: :$\map {\Gamma'} 1 = -\gamma$ where: :$\map {\Gamma'} 1$ denotes the derivative of the Gamma function evaluated at $1$ :$\gamma$ denotes the Euler-Mascheroni constant.
Stirling's Formula for Gamma Function
Let $\Gamma$ denote the Gamma function. Then: :$\Gamma \left({x + 1}\right) = \sqrt {2 \pi x} \, x^x e^{-x} \left({1 + \dfrac 1 {12 x} + \dfrac 1 {288 x^2} - \dfrac {139} {51 \, 480 x^3} + \cdots}\right)$
Beta Function as Integral of Power of t over Power of t plus 1
:$\displaystyle \map \Beta {x, y} = \int_{\mathop \to 0}^{\mathop \to \infty} \frac {t^{x - 1} } {\paren {1 + t}^{x + y} } \rd t$ where $\Beta$ denotes the Beta function.
Beta Function as Integral of Power of t by Power of 1 minus t over Power of r plus t
:$\displaystyle \Beta \left({x, y}\right) := r^y \left({r + 1}\right)^x \int_{\mathop \to 0}^{\mathop \to 1} \frac {t^{x - 1} \left({1 - t}\right)^{y - 1} } {\left({r + t}\right)^{x + y} } \rd t$ where $\Beta$ denotes the Beta function.
Separation of Variables/General Result
Suppose a first order ordinary differential equation can be expressible in this form: :$\map {g_1} x \map {h_1} y + \map {g_2} x \map {h_2} y \dfrac {\d y} {\d x} = 0$ Then the equation is said to '''have separable variables''', or '''be separable'''. Its general solution is found by solving the integration: :$\displaystyle \int \frac {\map {g_1} x} {\map {g_2} x} \rd x + \int \frac {\map {h_2} y} {\map {h_1} y} \rd y = C$
Sum of Arithmetic-Geometric Sequence
Let $\sequence {a_k}$ be an arithmetic-geometric sequence defined as: :$a_k = \paren {a + k d} r^k$ for $k = 0, 1, 2, \ldots, n - 1$ Then its closed-form expression is: :$\displaystyle \sum_{k \mathop = 0}^{n - 1} \paren {a + k d} r^k = \frac {a \paren {1 - r^n} } {1 - r} + \frac {r d \paren {1 - n r^{n - 1} + \paren {n - 1} r^n} } {\paren {1 - r}^2}$
Reduction Formula for Integral of Power of Cosine
Let $n \in \Z_{> 0}$ be a (strictly) positive integer. Then: :$\displaystyle \int \cos^n x \rd x = \dfrac {\cos^{n - 1} x \sin x} n + \dfrac {n - 1} n \int \cos^{n - 2} x \rd x$ is a reduction formula for $\displaystyle \int \cos^n x \rd x$.
Primitives of Rational Functions involving Power of a x + b
This page gathers together the primitives of some rational functions involving a general power of $a x + b$.
Primitives of Functions involving Root of a x + b
This page gathers together the primitives of some rational functions involving $\sqrt {a x + b}$.
Primitives of Functions involving Power of Root of a x + b
This page gathers together the primitives of some rational functions involving $\sqrt {a x + b}$.
Primitives of Functions involving a x + b and p x + q
This page gathers together the primitives of some rational functions involving $a x + b$ and $p x + q$.
Primitives of Functions involving Root of a x + b and p x + q
This page gathers together the primitives of some functions involving $\sqrt{a x + b}$ and $p x + q$.
Primitives of Functions involving Root of a x + b and Root of p x + q
This page gathers together the primitives of some functions involving $\sqrt{a x + b}$ and $\sqrt{p x + q}$.
Primitives involving x squared plus a squared
This page gathers together the primitives of some expressions involving $x^2 + a^2$.
Primitives involving x squared plus a squared squared
This page gathers together the primitives of some expressions involving $\left({x^2 + a^2}\right)^2$.
Primitives involving Power of x squared plus a squared
This page gathers together the primitives of some expressions involving $\left({x^2 + a^2}\right)^n$.
Primitives involving x squared minus a squared
This page gathers together the primitives of some expressions involving $x^2 - a^2$.
Primitives involving x squared minus a squared squared
This page gathers together the primitives of some expressions involving $\left({x^2 - a^2}\right)^2$.
Primitives involving Power of x squared minus a squared
This page gathers together the primitives of some expressions involving $\left({x^2 - a^2}\right)^n$.
Primitives involving a squared minus x squared
This page gathers together the primitives of some expressions involving $a^2 - x^2$.
Primitives involving a squared minus x squared squared
This page gathers together the primitives of some expressions involving $\left({a^2 - x^2}\right)^2$.
Primitives involving Power of a squared minus x squared
This page gathers together the primitives of some expressions involving $\left({x^2 - a^2}\right)^n$.
Primitives involving Root of x squared plus a squared
This page gathers together the primitives of some expressions involving $\sqrt{x^2 + a^2}$.
Primitives involving Root of x squared plus a squared cubed
This page gathers together the primitives of some expressions involving $\paren {\sqrt {x^2 + a^2} }^3$.
Primitives involving Root of x squared minus a squared
This page gathers together the primitives of some expressions involving $\sqrt{x^2 - a^2}$.
Primitives involving Root of x squared minus a squared cubed
This page gathers together the primitives of some expressions involving $\left({\sqrt{x^2 - a^2} }\right)^3$.
Primitives involving Root of a squared minus x squared
This page gathers together the primitives of some expressions involving $\sqrt {a^2 - x^2}$.
Primitives involving Root of a squared minus x squared cubed
This page gathers together the primitives of some expressions involving $\left({\sqrt{a^2 - x^2} }\right)^3$.
Primitives involving a x squared plus b x plus c
This page gathers together the primitives of some expressions involving $a x^2 + b x + c$.
Primitive of Reciprocal of Root of a x squared plus b x plus c
Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {\d x} {\sqrt {a x^2 + b x + c} } = \begin{cases} \dfrac 1 {\sqrt a} \map \ln {2 \sqrt a \sqrt {a x^2 + b x + c} + 2 a x + b} + C & : b^2 - 4 a c > 0 \\ \dfrac 1 {\sqrt a} \map {\sinh^{-1} } {\dfrac {2 a x + b} {\sqrt {4 a c - b^2} } } + C & : b^2 - 4 a c < 0 \\ \dfrac 1 {\sqrt a} \ln \size {2 a x + b} + C & : b^2 - 4 a c = 0 \end{cases}$
Primitive of Reciprocal of Root 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: :$\displaystyle \int \frac {\d x} {\sqrt {a x^2 + b x + c} } = \frac 1 {\sqrt a} \sinh^{-1} \paren {\dfrac {2 a x + b} {\sqrt {4 a c - b^2} } } + C$
Primitive of Reciprocal of Root 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: :$\displaystyle \int \frac {\d x} {\sqrt {a x^2 + b x + c} } = \frac 1 {\sqrt a} \map \ln {2 \sqrt a \sqrt {a x^2 + b x + c} + 2 a x + b} + C$
Primitive of Reciprocal of Root 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: :$\displaystyle \int \frac {\mathrm d x} {\sqrt {a x^2 + b x + c} } = \frac 1 {\sqrt a} \ln \left\vert{2 a x + b}\right\vert + C$
Primitive of Reciprocal of a x squared plus b x plus c/b equal to 0/Proof 1
Let $a \in \R_{\ne 0}$. Let $b = 0$. Then: :$\displaystyle \int \frac {\d x} {a x^2 + b x + c} = \begin {cases} \dfrac 1 {\sqrt {a c} } \map \arctan {x \sqrt {\dfrac a c} } + C & : a c > 0 \\ \dfrac 1 {2 \sqrt {-a c} } \ln \size {\dfrac {a x - \sqrt {-a c} } {a x + \sqrt {-a c} } } + C & : a c < 0 \\ \dfrac {-1} {a x} + C & : c = 0 \end {cases}$
Primitive of Reciprocal of x by Root of a x squared plus b x plus c
Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {\d x} {x \sqrt {a x^2 + b x + c} } = \begin{cases} \dfrac {-1} {\sqrt c} \ln \left({\dfrac {2 \sqrt c \sqrt {a x^2 + b x + c} + b x + 2 c} x}\right) & : b^2 - 4 a c > 0 \\ \dfrac {-1} {\sqrt c} \sinh^{-1} \left({\dfrac {b x + 2 c} {\left\vert{x}\right\vert \sqrt {4 a c - b^2} } }\right) & : b^2 - 4 a c < 0 \\ \dfrac {-1} {\sqrt c} \ln \left\vert{\dfrac {2 c} x + b}\right\vert + C & : b^2 - 4 a c = 0 \end{cases}$
Primitive of Reciprocal of x squared by Root of a x squared plus b x plus c
Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {\d x} {x^2 \sqrt {a x^2 + b x + c} } = -\frac {\sqrt {a x^2 + b x + c} } {c x} - \frac b {2 c} \int \frac {\d x} {x \sqrt {a x^2 + b x + c} }$
Primitive of Root of a x squared plus b x plus c
Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \sqrt {a x^2 + b x + c} \rd x = \frac {\paren {2 a x + b} \sqrt {a x^2 + b x + c} } {4 a} + \frac {4 a c - b^2} {8 a} \int \frac {\d x} {\sqrt {a x^2 + b x + c} }$
Primitive of x by Root of a x squared plus b x plus c
Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int x \sqrt {a x^2 + b x + c} \ \mathrm d x = \frac {\left({\sqrt {a x^2 + b x + c} }\right)^3} {3 a} - \frac {b \left({2 a x + b}\right) \sqrt {a x^2 + b x + c} } {8 a^2} - \frac {b \left({4 a c - b^2}\right)} {16 a^2} \int \frac {\mathrm d x} {\sqrt {a x^2 + b x + c} }$
Primitive of x squared by Root of a x squared plus b x plus c
Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int x^2 \sqrt {a x^2 + b x + c} \rd x = \frac {6 a x - 5 b} {24 a^2} \paren {\sqrt {a x^2 + b x + c} }^3 + \frac {5 b^2 - 4 a c} {16 a^2} \int \sqrt {a x^2 + b x + c} \rd x$
Primitive of Root of a x squared plus b x plus c over x
Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {\sqrt {a x^2 + b x + c} } x \rd x = \sqrt {a x^2 + b x + c} + \frac b 2 \int \frac {\d x} {\sqrt {a x^2 + b x + c} } + c \int \frac {\d x} {x \sqrt {a x^2 + b x + c} }$
Primitive of Root of a x squared plus b x plus c over x squared
Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {\sqrt {a x^2 + b x + c} } {x^2} \ \mathrm d x = \frac {-\sqrt {a x^2 + b x + c} } x + a \int \frac {\mathrm d x} {\sqrt {a x^2 + b x + c} } + \frac b 2 \int \frac {\mathrm d x} {x \sqrt {a x^2 + b x + c} }$
Primitive of Reciprocal of Cube of Root of a x squared plus b x plus c
Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {\mathrm d x} {\left({\sqrt {a x^2 + b x + c} }\right)^3} = \frac {2 \left({2 a x + b}\right)} {\left({4 a c - b^2}\right) \sqrt {a x^2 + b x + c} } + C$
Primitive of x over Cube of Root of a x squared plus b x plus c
Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {x \ \mathrm d x} {\left({\sqrt {a x^2 + b x + c} }\right)^3} = \frac {2 \left({b x + 2 c}\right)} {\left({b^2 - 4 a c}\right) \sqrt {a x^2 + b x + c} }$
Primitive of Reciprocal of x by Cube of Root of a x squared plus b x plus c
Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {\mathrm d x} {x \left({\sqrt {a x^2 + b x + c} }\right)^3} = \frac 1 {c \sqrt {a x^2 + b x + c} } + \frac 1 c \int \frac {\mathrm d x} {x \sqrt {a x^2 + b x + c} } - \frac b {2 c} \int \frac {\mathrm d x} {\left({\sqrt {a x^2 + b x + c} }\right)^3}$
Primitive of x by Half Integer Power of a x squared plus b x plus c
Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int x \left({a x^2 + b x + c}\right)^{n + \frac 1 2} \ \mathrm d x = \frac {\left({a x^2 + b x + c}\right)^{n + \frac 3 2} } {a \left({2 n + 3}\right)} - \frac b {2 a} \int \left({a x^2 + b x + c}\right)^{n + \frac 1 2} \ \mathrm d x$
Primitive of x over x cubed plus a cubed
:$\displaystyle \int \frac {x \rd x} {x^3 + a^3} = \frac 1 {6 a} \map \ln {\frac {x^2 - a x + a^2} {\paren {x + a}^2} } + \frac 1 {a \sqrt 3} \arctan \frac {2 x - a} {a \sqrt 3}$
Primitive of Reciprocal of x cubed plus a cubed
:$\displaystyle \int \frac {\d x} {x^3 + a^3} = \frac 1 {6 a^2} \ln \size {\frac {\paren {x + a}^2} {x^2 - a x + a^2} } + \frac 1 {a^2 \sqrt 3} \arctan \frac {2 x - a} {a \sqrt 3}$
Primitive of Reciprocal of x by x cubed plus a cubed
:$\displaystyle \int \frac {\mathrm d x} {x \left({x^3 + a^3}\right)} = \frac 1 {3 a^3} \ln \left\vert{\frac {x^3} {x^3 + a^3} }\right\vert + C$
Primitive of Reciprocal of x cubed plus a cubed squared
:$\displaystyle \int \frac {\mathrm d x} {\left({x^3 + a^3}\right)^2} = \frac x {3 a^3 \left({x^3 + a^3}\right)} + \frac 1 {9 a^5} \ln \left({\frac {\left({x + a}\right)^2} {x^2 - a x + a^2} }\right) + \frac 2 {3 a^5 \sqrt 3} \arctan \frac {2 x - a} {a \sqrt 3}$
Primitive of x squared over x cubed plus a cubed squared
:$\displaystyle \int \frac {x^2 \ \mathrm d x} {\left({x^3 + a^3}\right)^2} = \frac {-1} {3 \left({x^3 + a^3}\right)} + C$
Primitive of Reciprocal of x by x cubed plus a cubed squared
:$\displaystyle \int \frac {\mathrm d x} {x \left({x^3 + a^3}\right)^2} = \frac 1 {3 a^3 \left({x^3 + a^3}\right)} + \frac 1 {3 a^6} \ln \left\vert{\frac {x^3} {x^3 + a^3} }\right\vert$
Primitive of Power of x over x cubed plus a cubed
:$\displaystyle \int \frac {x^m \rd x} {x^3 + a^3} = \frac {x^{m - 2} } {m - 2} - a^3 \int \frac {x^{m - 3} \rd x} {x^3 + a^3}$
Primitive of Reciprocal of x fourth plus a fourth
:$\displaystyle \int \frac {\d x} {x^4 + a^4} = \frac 1 {4 a^3 \sqrt 2} \map \ln {\frac {x^2 + a x \sqrt 2 + a^2} {x^2 - a x \sqrt 2 + a^2} } - \frac 1 {2 a^3 \sqrt 2} \paren {\map \arctan {1 - \frac {x \sqrt 2} a} - \map \arctan {1 + \frac {x \sqrt 2} a} }$
Primitive of x over x fourth plus a fourth
:$\displaystyle \int \frac {x \rd x} {x^4 + a^4} = \frac 1 {2 a^2} \arctan \frac {x^2} {a^2}$
Primitive of x squared over x fourth plus a fourth
:$\displaystyle \int \frac {x^2 \rd x} {x^4 + a^4} = \frac 1 {4 a \sqrt 2} \map \ln {\frac {x^2 - a x \sqrt 2 + a^2} {x^2 + a x \sqrt 2 + a^2} } - \frac 1 {2 a \sqrt 2} \paren {\map \arctan {1 - \frac {x \sqrt 2} a} - \map \arctan {1 + \frac {x \sqrt 2} a} }$
Primitive of x cubed over x fourth plus a fourth
:$\displaystyle \int \frac {x^3 \ \mathrm d x} {x^4 + a^4} = \frac {\ln \left({x^4 + a^4}\right)} 4$
Primitive of Reciprocal of x by x fourth plus a fourth
:$\displaystyle \int \frac {\mathrm d x} {x \left({x^4 + a^4}\right)} = \frac 1 {4 a^4} \ln \left({\frac {x^4} {x^4 + a^4} }\right)$
Primitive of Reciprocal of x cubed by x fourth plus a fourth
:$\displaystyle \int \frac {\mathrm d x} {x^3 \left({x^4 + a^4}\right)} = \frac {-1} {2 a^4 x^2} - \frac 1 {2 a^6} \arctan \frac {x^2} {a^2}$
Primitive of x over x fourth minus a fourth
:$\displaystyle \int \frac {x \ \mathrm d x} {x^4 - a^4} = \frac 1 {4 a^2} \ln \left\vert{\frac {x^2 - a^2} {x^2 + a^2} }\right\vert + C$
Primitive of x squared over x fourth minus a fourth
:$\displaystyle \int \frac {x^2 \rd x} {x^4 - a^4} = \frac 1 {4 a} \ln \size {\frac {x - a} {x + a} } + \frac 1 {2 a} \arctan \frac x a + C$
Primitive of x cubed over x fourth minus a fourth
:$\displaystyle \int \frac {x^3 \ \mathrm d x} {x^4 - a^4} = \frac {\ln \left\vert{x^4 - a^4}\right\vert} 4 + C$
Primitive of Reciprocal of x squared by x fourth minus a fourth
:$\displaystyle \int \frac {\mathrm d x} {x^2 \left({x^4 - a^4}\right)} = \frac 1 {a^4 x} + \frac 1 {4 a^5} \ln \left\vert{\frac {x - a} {x + a} }\right\vert + \frac 1 {2 a^5} \arctan \frac x a + C$
Primitive of Reciprocal of x cubed by x fourth minus a fourth
:$\displaystyle \int \frac {\mathrm d x} {x^3 \left({x^4 - a^4}\right)} = \frac 1 {2 a^4 x^2} + \frac 1 {4 a^6} \ln \left\vert{\frac {x^2 - a^2} {x^2 + a^2} }\right\vert + C$
Primitive of Reciprocal of x by Power of x plus Power of a
:$\displaystyle \int \frac {\mathrm d x} {x \left({x^n + a^n}\right)} = \frac 1 {n a^n} \ln \left\vert{\frac {x^n} {x^n + a^n} }\right\vert + C$
Absolute Value of Even Power
Let $x \in \R$ be a real number. Let $n \in \Z$ be an integer. Then: :$\left\vert{x^{2 n} }\right\vert = x^{2 n}$
Primitive of Power of x less one over Power of x plus Power of a
:$\displaystyle \int \frac {x^{n - 1} \ \mathrm d x} {x^n + a^n} = \frac 1 n \ln \left\vert{x^n + a^n}\right\vert + C$
Primitive of x cubed over x fourth plus a fourth/Proof 1
:$\displaystyle \int \frac {x^3 \ \mathrm d x} {x^4 + a^4} = \frac {\ln \left({x^4 + a^4}\right)} 4$
Primitive of x cubed over x fourth plus a fourth/Proof 2
:$\displaystyle \int \frac {x^3 \ \mathrm d x} {x^4 + a^4} = \frac {\ln \left({x^4 + a^4}\right)} 4$
Primitive of Reciprocal of Power of x by Power of Power of x plus Power of a
:$\displaystyle \int \frac {\d x} {x^m \ \left({x^n + a^n}\right)^r} = \frac 1 {a^n} \int \frac {\d x} {x^m \left({x^n + a^n}\right)^{r - 1} } - \frac 1 {a^n} \int \frac {\d x} {x^{m - n} \left({x^n + a^n}\right)^r}$
Primitive of Reciprocal of x by Root of Power of x plus Power of a
:$\displaystyle \int \frac {\d x} {x \sqrt {x^n + a^n} } = \frac 1 {n \sqrt {a^n} } \ln \size {\frac {\sqrt {x^n + a^n} - \sqrt {a^n} } {\sqrt {x^n + a^n} + \sqrt {a^n} } } + C$
Primitive of Reciprocal of x by Power of x minus Power of a
:$\displaystyle \int \frac {\d x} {x \paren {x^n - a^n} } = \frac 1 {n a^n} \ln \size {\frac {x^n - a^n} {x^n} } + C$
Primitive of Power of x less one over Power of x minus Power of a
:$\displaystyle \int \frac {x^{n - 1} \rd x} {x^n - a^n} = \frac 1 n \ln \size {x^n - a^n} + C$
Primitive of x by Sine of a x
:$\displaystyle \int x \sin a x \rd x = \frac {\sin a x} {a^2} - \frac {x \cos a x} a + C$
Primitive of x squared by Sine of a x
:$\displaystyle \int x^2 \sin a x \rd x = \frac {2 x \sin a x} {a^2} + \paren {\frac 2 {a^3} - \frac {x^2} a} \cos a x + C$
Primitive of Sine of a x over x squared
:$\displaystyle \int \frac {\sin a x \ \mathrm d x} {x^2} = -\frac {\sin a x} x + a \int \frac {\cos a x \ \mathrm d x} x$
Primitive of x over Sine of a x
{{begin-eqn}} {{eqn | l = \int \frac {x \rd x} {\sin a x}       | r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + C       | c =  }} {{eqn | r = \frac 1 {a^2} \paren {a x + \frac {\paren {a x}^3} {18} + \frac {7 \paren {a x}^5} {1800} \cdots} + C       | c =  }} {{end-eqn}}
Primitive of Square of Sine of a x
:$\displaystyle \int \sin^2 a x \rd x = \frac x 2 - \frac {\sin 2 a x} {4 a} + C$
Primitive of x by Square of Sine of a x
:$\ds \int x \sin^2 a x \rd x = \frac {x^2} 4 - \frac {x \sin 2 a x} {4 a} - \frac {\cos 2 a x} {8 a^2} + C$
Primitive of Cube of Sine of a x
:$\displaystyle \int \sin^3 a x \rd x = -\frac {\cos a x} a + \frac {\cos^3 a x} {3 a} + C$
Primitive of Square of Cosecant of a x
:$\displaystyle \int \csc^2 a x \ \mathrm d x = -\frac {\cot a x} a + C$
Primitive of Reciprocal of Square of Sine of a x
:$\displaystyle \int \frac {\mathrm d x} {\sin^2 a x} = \frac {-\cot a x} a + C$
Primitive of Reciprocal of Cube of Sine of a x
:$\displaystyle \int \frac {\mathrm d x} {\sin^3 a x} = \frac {-\cos a x} {2 a \sin^2 a x} + \frac 1 {2 a} \ln \left\vert{\tan \frac {a x} 2}\right\vert + C$
Primitive of Sine of p x by Sine of q x
For $p \ne q$: :$\displaystyle \int \sin p x \sin q x \rd x = \frac {\map \sin {p - q} x} {2 \paren {p - q} } - \frac {\map \sin {p + q} x} {2 \paren {p + q} } + C$
Primitive of Reciprocal of 1 minus Sine of a x
:$\displaystyle \int \frac {\d x} {1 - \sin a x} = \frac 1 a \tan \left({\frac \pi 4 + \frac {a x} 2}\right) + C$
Primitive of x over 1 minus Sine of a x
:$\ds \int \frac {x \rd x} {1 - \sin a x} = \frac x a \map \tan {\frac \pi 4 + \frac {a x} 2} + \frac 2 {a^2} \ln \size {\map \sin {\frac \pi 4 - \frac {a x} 2} } + C$
Primitive of Reciprocal of 1 plus Sine of a x
:$\displaystyle \int \frac {\d x} {1 + \sin a x} = -\frac 1 a \map \tan {\frac \pi 4 - \frac {a x} 2} + C$
Primitive of x over 1 plus Sine of a x
:$\ds \int \frac {x \rd x} {1 + \sin a x} = -\frac x a \map \tan {\frac \pi 4 - \frac {a x} 2} + \frac 2 {a^2} \ln \size {\map \sin {\frac \pi 4 + \frac {a x} 2} } + C$
Reciprocal of One Minus Sine
:$\dfrac 1 {1 - \sin x} = \dfrac 1 2 \map {\sec^2} {\dfrac \pi 4 + \dfrac x 2}$
Reciprocal of One Plus Sine
:$\dfrac 1 {1 + \sin x} = \dfrac 1 2 \map {\sec^2} {\dfrac \pi 4 - \dfrac x 2}$
Power Series Expansion for Cosecant Function
The cosecant function has a Laurent series expansion: {{begin-eqn}} {{eqn | l = \csc x       | r = \sum_{n \mathop = 0}^\infty \dfrac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} \,  x^{2 n - 1} } {\paren {2 n}!}       | c =  }} {{eqn | r = \frac 1 x + \frac x 6 + \frac {7 x^3} {360} + \frac {31 x^5} {15 \, 120} + \cdots       | c =  }} {{end-eqn}} where $B_n$ denotes the Bernoulli numbers. This converges for $0 < \size x < \pi$.
Periodic Function plus Constant
Let $f: \mathbb F \to \mathbb F$ be a function, where $\mathbb F \in \left\{{\R, \C}\right\}$. Let $k \in \mathbb F$ be constant. Then $f$ is periodic with period $L$  $f + k$ is periodic with period $L$.
Sum of Two Odd Powers/Examples/Sum of Two Cubes
:$x^3 + y^3 = \paren {x + y} \paren {x^2 - x y + y^2}$
Primitive of Periodic Function
Let $f: \R \to \R$ be a real function. Let $F$ be a primitive of $f$ that is bounded on all of $\R$. Let $f$ be periodic with period $L$. Then $F$ is also periodic with period $L$.
Primitive of Reciprocal of x squared by x cubed plus a cubed/Lemma
:$\displaystyle \int \frac {\mathrm d x} {x^2 \left({x^3 + a^3}\right)} = \frac {-1} {a^3 x} - \frac 1 {a^3} \int \frac {x \ \mathrm d x} {\left({x^3 + a^3}\right)}$
Sum of Two Fourth Powers
:$x^4 + y^4 = \paren {x^2 + \sqrt 2 x y + y^2} \paren {x^2 - \sqrt 2 x y + y^2}$
One Plus Tangent Half Angle over One Minus Tangent Half Angle
:$\dfrac {1 + \tan \frac x 2} {1 - \tan \frac x 2} = \sec x + \tan x$
Primitive of Power of Cosecant of a x
:$\displaystyle \int \csc^n a x \ \mathrm d x = \frac{-\csc^{n - 2} a x \cot a x} {a \left({n - 1}\right)} + \frac {n - 2} {n - 1} \int \csc^{n - 2} a x \ \mathrm d x$
Primitive of Cosecant of a x/Tangent Form
:$\ds \int \csc a x \rd x = \frac 1 a \ln \size {\tan \frac {a x} 2} + C$ where $\tan \dfrac {a x} 2 \ne 0$.
Primitive of Reciprocal of Square of 1 plus Sine of a x
:$\displaystyle \int \frac {\mathrm d x} {\left({1 + \sin a x}\right)^2} = \frac {-1} {2a} \tan \left({\frac \pi 4 - \frac {a x} 2}\right) - \frac 1 {6 a} \tan^3 \left({\frac \pi 4 - \frac {a x} 2}\right) + C$
Primitive of Reciprocal of p plus q by Sine of a x
:$\displaystyle \int \frac {\d x} {p + q \sin a x} = \begin{cases}  \displaystyle \frac 2 {a \sqrt {p^2 - q^2} } \map \arctan {\frac {p \tan \dfrac {a x} 2 + q} {\sqrt {p^2 - q^2} } } + C & : q^2 - p^2 < 0 \\ \displaystyle \frac 1 {a \sqrt {q^2 - p^2} } \ln \size {\frac {p \tan \dfrac {a x} 2 + q - \sqrt {p^2 - q^2} } {p \tan \dfrac {a x} 2 + q + \sqrt {p^2 - q^2} } } + C & : q^2 - p^2 > 0 \\ \end{cases}$
Primitive of Reciprocal of square of p plus q by Sine of a x
:$\displaystyle \int \frac {\d x} {\paren {p + q \sin a x}^2} = \frac {q \cos a x} {a \paren {p^2 - q^2} \paren {p + q \sin a x} } + \frac p {p^2 - q^2} \int \frac {\d x} {p + q \sin a x}$
Primitive of Reciprocal of p squared plus square of q by Sine of a x
:$\displaystyle \int \frac {\d x} {p^2 + q^2 \sin^2 a x} = \frac 1 {a p \sqrt {p^2 + q^2} } \arctan \frac {\sqrt {p^2 + q^2} \tan a x} p + C$
Primitive of Sine of a x over Power of x
:$\displaystyle \int \frac {\sin a x} {x^n} \rd x = \frac {-\sin a x} {\paren {n - 1} x^{n - 1} } + \frac a {n - 1} \int \frac {\cos a x} {x^{n - 1} } \rd x$
Primitive of Reciprocal of Power of Sine of a x
:$\displaystyle \int \frac {\mathrm d x} {\sin^n a x} = \frac {- \cos a x} {a \left({n - 1}\right) \sin^{n - 1} a x} + \frac {n - 2} {n - 1} \int \frac {\mathrm d x} {\sin^{n - 2} a x}$
Primitive of x over Power of Sine of a x
:$\displaystyle \int \frac {x \rd x} {\sin^n a x} = \frac {-x \cos a x} {a \paren {n - 1} \sin^{n - 1} a x} - \frac 1 {a^2 \paren {n - 1} \paren {n - 2} \sin^{n - 2} a x} + \frac {n - 2} {n - 1} \int \frac {x \rd x} {\sin^{n - 2} a x}$
Primitive of x by Cosine of a x
:$\displaystyle \int x \cos a x \rd x = \frac {\cos a x} {a^2} + \frac {x \sin a x} a + C$
Primitive of x squared by Cosine of a x
:$\displaystyle \int x^2 \cos a x \rd x = \frac {2 x \cos a x} {a^2} + \paren {\frac {x^2} a - \frac 2 {a^3} } \sin a x + C$
Primitive of x cubed by Cosine of a x
:$\displaystyle \int x^3 \map \cos {a x} \rd x = \paren {\frac {3 x^2} {a^2} - \frac 6 {a^4} } \cos a x + \paren {\frac {x^3} a - \frac {6 x} {a^3} } \sin a x + C$
Primitive of Reciprocal of Cosine of a x/Logarithm of Tangent Form
:$\ds \int \frac {\d x} {\cos a x} = \frac 1 a \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C$
Primitive of Square of Cosine of a x
:$\displaystyle \int \cos^2 a x \, \mathrm d x = \frac x 2 + \frac {\sin 2 a x} {4 a} + C$
Primitive of x by Square of Cosine of a x
:$\displaystyle \int x \cos^2 a x \ \mathrm d x = \frac {x^2} 4 + \frac {x \sin 2 a x} {4 a} + \frac {\cos 2 a x} {8 a^2} + C$
Primitive of Fourth Power of Cosine of a x
:$\displaystyle \int \cos^4 a x \ \mathrm d x = \frac {3 x} 8 + \frac {\sin 2 a x} {4 a} + \frac {\sin 4 a x} {32 a} + C$
Primitive of Reciprocal of Square of Cosine of a x
:$\displaystyle \int \frac {\mathrm d x} {\cos^2 a x} = \frac {\tan a x} a + C$