hcju/proofwikips huggingface dataset:
Comparison Test for Improper Integral
Let $I = \openint a b$ be an open real interval. Let $\phi$ be a real function which is continuous on $I$ and also non-negative on $I$. Let $f$ be a real function which is continuous on $I$. Let $f$ satisfy: :$\forall x \in I: \size {\map f x} \le \map \phi x$ If the improper integral of $\phi$ over $I$ exists, then so does that of $f$.
Beta Function is Continuous and Positive on Positive Reals
Let $x, y \in \R$ be real numbers. Let $\map \Beta {x, y}$ be the Beta function: :$\ds \map \Beta {x, y} := \int_{\mathop \to 0}^{\mathop \to 1} t^{x - 1} \paren {1 - t}^{y - 1} \rd t$ Let $y \in \R_{>0}$ be given. Then $\map \Beta {x, y}$ is a positive and continuous function of $x$ on $\R_{>0}$.
Logarithm of Beta Function is Convex on Positive Reals
Let $x, y \in \R$ be real numbers. Let $\Beta \left({x, y}\right)$ be the Beta function: :$\displaystyle \Beta \left({x, y}\right) := \int_{\mathop \to 0}^{\mathop \to 1} t^{x - 1} \left({1 - t}\right)^{y - 1} \ \mathrm d t$ Let $y \in \R_{>0}$ be given. Then $\ln \left({\Beta \left({x, y}\right)}\right)$ is a convex function of $x$ on $\R_{>0}$.
Gamma Function of One Half
:$\map \Gamma {\dfrac 1 2} = \sqrt \pi$ Its decimal expansion starts: :$\map \Gamma {\dfrac 1 2} = 1 \cdotp 77245 \, 38509 \, 05516 \, 02729 \, 81674 \, 83341 \, 14518 \, 27975 \ldots$
Sum of Reciprocals of Primes is Divergent/Proof 2
The series: :$\displaystyle \sum_{p \mathop \in \Bbb P} \frac 1 p$ where: :$\Bbb P$ is the set of all prime numbers is divergent.
Sum of Reciprocals of Primes is Divergent/Lemma
Let $C \in \R_{>0}$ be a (strictly) positive real number. Then: :$\displaystyle \lim_{n \mathop \to \infty} \paren {\map \ln {\ln n} - C} = + \infty$
Strictly Increasing Infinite Sequence of Integers is Cofinal in Natural Numbers
Let $S = \left\langle{x_n}\right\rangle$ be an infinite sequence of integers which is strictly increasing.   Then $S$ is a cofinal subset of $\left({\Z, \le}\right)$ where $\le$ is the usual ordering on the integers.
Product of Even and Odd Functions
Let $\mathcal O$ be an odd real function defined on some symmetric set $S$. Let $\mathcal E$ be an even real function defined on some symmetric set $S'$. Let $\mathcal O\mathcal E$ be their pointwise product, defined on the intersection of the domains of $\mathcal O$ and $\mathcal E$. Then $\mathcal O\mathcal E$ is odd.  That is: :$\forall x \in S \cap S': \left({\mathcal O\mathcal E}\right)\left({-x}\right) = - \left({\mathcal O\mathcal E}\right)\left({x}\right)$.
Millin Series
The '''Millin series''' is the series defined as: :$\displaystyle \sum_{n \mathop = 0}^\infty \frac 1 {F_{2^n}} = \frac {7 - \sqrt 5} 2$
Odd Bernoulli Numbers Vanish
Let $B_n$ denote the $n$th Bernoulli Number. Then: :$B_{2n + 1} = 0$ for $n \ge 1$.
Gauss Multiplication Formula
Let $\Gamma$ denote the Gamma Function. Then: :$\displaystyle \forall z \notin \set {-\frac m n: m \in \N}: \prod_{k \mathop = 0}^{n - 1} \map \Gamma {z + \frac k n} = \paren {2 \pi}^{\paren {n - 1} / 2} n^{1/2 - n z} \map \Gamma {n z}$ where $\N$ denotes the natural numbers.
Dirichlet Integral
:$\displaystyle \int_0^\infty \frac {\sin x} x \rd x = \frac \pi 2$
Riemann Zeta Function and Prime Counting Function
For $\map \Re s > 1$: :$\displaystyle \log \map \zeta s = s \int_0^{\mathop \to \infty} \frac {\map \pi x} {x \paren {x^s - 1} } \rd x$ where: :$\zeta$ denotes the Riemann Zeta Function :$\pi$ denotes the Prime-Counting Function.
Faulhaber's Formula
Let $n$ and $p$ be positive integers. Then: :$\displaystyle \sum_{k \mathop = 1}^n k^p = \frac 1 {p + 1} \sum_{i \mathop = 0}^p \paren {-1}^i \binom {p + 1} i B_i n^{p + 1 - i}$ where $B_n$ denotes the $n$th Bernoulli number.
Complex Exponential Tends to Zero
Let $\exp z$ be the complex exponential function. Then: :$\displaystyle \lim_{\map \Re z \mathop \to +\infty} e^{-z} = 0$ where $\map \Re z$ denotes the real part of $z$.
Modulus of Exponential is Exponential of Real Part
Let $z \in \C$ be a complex number. Let $\exp z$ denote the complex exponential function. Let $\cmod {\, \cdot \,}$ denote the complex modulus Then: :$\cmod {\exp z} = \map \exp {\Re z}$ where $\Re z$ denotes the real part of $z$.
Laplace Transform of Exponential/Real Argument
Let $\laptrans f$ denote the Laplace transform of a function $f$. Let $e^x$ be the real exponential. Then: :$\map {\laptrans {e^{a t} } } s = \dfrac 1 {s - a}$ where $a \in \R$ is constant, and $\map \Re s > \map \Re a$.
Laplace Transform of Cosine
Let $\cos$ be the real cosine function. Let $\laptrans f$ denote the Laplace transform of the real function $f$. Then: :$\laptrans {\cos a t} = \dfrac s {s^2 + a^2}$ where $a \in \R_{>0}$ is constant, and $\map \Re s > a$.
Laplace Transform of Sine
Let $\sin$ denote the real sine function. Let $\laptrans f$ denote the Laplace transform of a real function $f$. Then: :$\laptrans {\sin at} = \dfrac a {s^2 + a^2}$ where $a \in \R_{>0}$ is constant, and $\map \Re s > 0$.
Sum Rule for Derivatives/General Result
Let $f_1 \left({x}\right), f_2 \left({x}\right), \ldots, f_n \left({x}\right)$ be real functions all differentiable. Then for all $n \in \N_{>0}$: : $\displaystyle D_x \left({\sum_{i \mathop = 1}^n f_i \left({x}\right)}\right) = \sum_{i \mathop = 1}^n D_x \left({f_i \left({x}\right)}\right)$
Product Rule for Derivatives/General Result
Let $f_1 \left({x}\right), f_2 \left({x}\right), \ldots, f_n \left({x}\right)$ be real functions differentiable on the open interval $I$. then: :$\forall x \in I: \displaystyle D_x \left({\prod_{i \mathop = 1}^n f_i \left({x}\right)}\right) = \sum_{i \mathop = 1}^n \left({D_x \left({f_i \left({x}\right)}\right) \prod_{j \mathop \ne i} f_j \left({x}\right)}\right)$
Derivative of Arccotangent Function
:$\dfrac {\map \d {\arccot x} } {\d x} = \dfrac {-1} {1 + x^2}$
Asymptotic Formula for Bernoulli Numbers
The Bernoulli numbers with even index can be approximated by the asymptotic formula: :$B_{2 n} \sim \paren {-1}^{n + 1} 4 \sqrt {\pi n} \paren {\dfrac n {\pi e} }^{2 n}$ where: :$B_n$ denotes the $n$th Bernoulli number :$\sim$ denotes asymptotically equal.
Cotangent Minus Tangent
:$\cot x - \tan x = 2 \cot 2 x$
Derivative of Hyperbolic Cotangent Function
:$\map {\dfrac \d {\d x} } {\coth u} = -\csch^2 u \dfrac {\d u} {\d x}$
Derivative of Hyperbolic Cosecant Function
:$\map {\dfrac \d {\d x} } {\csch u} = -\csch u \coth u \dfrac {\d u} {\d x}$
Derivative of Inverse Hyperbolic Sine
:$\map {\dfrac \d {\d x} } {\sinh^{-1} x} = \dfrac 1 {\sqrt {x^2 + 1} }$
Derivative of Inverse Hyperbolic Cosine
:$\map {\dfrac \d {\d x} } {\cosh^{-1} x} = \dfrac 1 {\sqrt {x^2 - 1} }$
Derivative of Inverse Hyperbolic Tangent
:$\map {\dfrac \d {\d x} } {\tanh^{-1} x} = \dfrac 1 {1 - x^2}$
Derivative of Inverse Hyperbolic Cotangent
:$\map {\dfrac \d {\d x} } {\coth^{-1} x} = \dfrac {-1} {x^2 - 1}$
Derivative of Inverse Hyperbolic Secant
:$\map {\dfrac \d {\d x} } {\sech^{-1} x} = \dfrac {-1} {x \sqrt{1 - x^2} }$
Derivative of Inverse Hyperbolic Cosecant
:$\map {\dfrac \d {\d x} } {\csch^{-1} x} = \dfrac {-1} {\size x \sqrt {1 + x^2} }$
Sum of Cosecant and Cotangent
:$\displaystyle \csc x + \cot x = \cot {\frac x 2}$
Leibniz's Rule/One Variable/Third Derivative
Let $f$ and $g$ be real functions defined on the open interval $I$. Let $x \in I$ be a point in $I$ at which both $f$ and $g$ are thrice differentiable. Then: :$\paren {\map f x \, \map g x}''' = \map f x \, \map {g'''} x + 3 \map {f'} x \, \map {g''} x + 3 \map {f''} x \, \map {g'} x + \map {f'''} x \, \map g x$
Power Series Expansion for Tangent Function
The tangent function has a Taylor series expansion: {{begin-eqn}} {{eqn | l = \tan x       | r = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!}       | c =  }} {{eqn | r = x + \frac {x^3} 3 + \frac {2 x^5} {15} + \frac {17 x^7} {315} + \cdots       | c =  }} {{end-eqn}} where $B_{2 n}$ denotes the Bernoulli numbers. This converges for $\size x < \dfrac \pi 2$.
Partial Differentiation Operator is Commutative for Continuous Functions
Let $\map f {x, y}$ be a function of the two independent variables $x$ and $y$. Let $\map f {x, y}$ be continuous. Let the partial deriviatives of $f$ also be continuous. Then: :$\dfrac {\partial^2 f} {\partial x \partial y} = \dfrac {\partial^2 f} {\partial y \partial x}$
Primitive of Constant Multiple of Function
Let $f$ be a real function which is integrable. Let $c$ be a constant. Then: :$\displaystyle \int c \map f x \rd x = c \int \map f x \rd x$
Primitive of Pointwise Sum of Functions
Let $f_1, f_2, \ldots, f_n$ be real functions which are integrable. Then: :$\ds \int \map {\paren {f_1 \pm f_2 \pm \, \cdots \pm f_n} } x \rd x = \int \map {f_1} x \rd x \pm \int \map {f_2} x \rd x \pm \, \cdots \pm \int \map {f_n} x \rd x$
Hurwitz's Theorem (Number Theory)
Let $\xi$ be an irrational number. Then there are infinitely many relatively prime integers $m, n \in \Z$ such that: :$\left|{\xi - \dfrac p q}\right| < \dfrac 1 {\sqrt 5 \, q^2}$
Primitive of Function of Constant Multiple
Let $f$ be a real function which is integrable. Let $c$ be a constant. Then: :$\displaystyle \int \map f {c x} \rd x = \frac 1 c \int \map f u \d u$ where $u = c x$.
Primitive of Composite Function
Let $f$ and $g$ be a real functions which are integrable. Let the composite function $g \circ f$ also be integrable. Then: {{begin-eqn}} {{eqn | l = \int \map {\paren {g \circ f} } x \rd x       | r = \int \map g u \frac {\d x} {\d u} \rd u       | c =  }} {{eqn | r = \int \frac {\map g u} {\map {f'} x} \rd u       | c =  }} {{end-eqn}} where $u = \map f x$.
Weierstrass Product Inequality
For $n \ge 1$: :$\displaystyle \prod_{i \mathop = 1}^n \left({1 - a_i}\right) \ge 1 - \sum_{i \mathop = 1}^n a_i$ where all of $a_i$ are in the closed interval $\left[{0 \,.\,.\,1 }\right]$.
Primitive of Power
Let $n \in \R: n \ne -1$. Then: :$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$ where $C$ is an arbitrary constant. That is: :$\dfrac {x^{n + 1} } {n + 1}$ is a primitive of $x^n$.
Sine Inequality
:$\size {\sin x} \le \size x$ for all $x \in \R$.
Jordan's Inequality
:$\dfrac 2 {\pi} x \le \sin x \le x$ for all $x$ in the interval $\left[{0 \,.\,.\, \dfrac {\pi} 2}\right]$
Cosine Inequality
:$1 - \dfrac {x^2} 2 \le \cos x$ for all $x \in \R$.
Laplace Transform of Constant Mapping
Let $a \in \R$ be a real constant. Let $f_a: \R \to \R$ or $\C$ be the constant mapping, defined as: :$\forall t \in \R: \map {f_a} t = a$ Let $\laptrans {f_a}$ be the Laplace transform of $f_a$. Then: :$\laptrans {\map {f_a} t} = \dfrac a s$ for $\map \Re s > a$.
Laplace Transform of Positive Integer Power
Let $\laptrans f$ denote the Laplace transform of a function $f$. Let $t^n: \R \to \R$ be $t$ to the $n$th power for some $n \in \N_{\ge 0}$. Then: :$\laptrans {t^n} = \dfrac {n!} { s^{n + 1} }$ for $\map \Re s > 0$.
Primitive of Function under its Derivative
Let $f$ be a real function which is integrable. Then: :$\ds \int \frac {\map {f'} x} {\map f x} \rd x = \ln \size {\map f x} + C$ where $C$ is an arbitrary constant.
Primitive of Exponential Function/General Result
Let $a \in \R_{>0}$ be a constant such that $a \ne 1$. Then: :$\ds \int a^x \rd x = \frac {a^x} {\ln a} + C$ where $C$ is an arbitrary constant.
Primitive of Tangent Function/Secant Form
:$\ds \int \tan x \rd x = \ln \size {\sec x} + C$ where $\sec x$ is defined.
Primitive of Secant Function/Tangent plus Angle Form
:$\ds \int \sec x \rd x = \ln \size {\map \tan {\frac x 2 + \frac \pi 4} } + C$
Exponential of x not less than 1+x
:$e^x \ge 1 + x$ for all $x \in \R$.
Primitive of Cosecant Function/Cosecant minus Cotangent Form
:$\ds \int \csc x \rd x = \ln \size {\csc x - \cot x} + C$ where $\csc x - \cot x \ne 0$.
Primitive of Cosecant Function/Tangent Form
:$\ds \int \csc x \rd x = \ln \size {\tan \frac x 2} + C$ where $\tan \dfrac x 2 \ne 0$.
Primitive of Square of Secant Function
:$\ds \int \sec^2 x \rd x = \tan x + C$ where $C$ is an arbitrary constant.
Primitive of Square of Cosecant Function
:$\ds \int \csc^2 x \rd x = -\cot x + C$ where $C$ is an arbitrary constant.
Primitive of Square of Tangent Function
:$\displaystyle \int \tan^2 x \rd x = \tan x - x + C$
Primitive of Square of Cotangent Function
:$\displaystyle \int \cot^2 x \rd x = -\cot x - x + C$ where $C$ is an arbitrary constant.
Primitive of Square of Sine Function
:$\ds \int \sin^2 x \rd x = \frac x 2 - \frac {\sin 2 x} 4 + C$ where $C$ is an arbitrary constant.
Primitive of Square of Cosine Function
:$\ds \int \cos^2 x \rd x = \frac x 2 + \frac {\sin 2 x} 4 + C$
Primitive of Product of Secant and Tangent
:$\ds \int \sec x \tan x \rd x = \sec x + C$ where $C$ is an arbitrary constant.
Primitive of Hyperbolic Sine Function
:$\ds \int \sinh x \rd x = \cosh x + C$
Primitive of Hyperbolic Cosine Function
:$\displaystyle \int \cosh x \rd x = \sinh x + C$
Primitive of Hyperbolic Tangent Function
:$\ds \int \tanh x \rd x = \map \ln {\cosh x} + C$
Primitive of Hyperbolic Cotangent Function
:$\ds \int \coth x \rd x = \ln \size {\sinh x} + C$ where $\sinh x \ne 0$.
Limit of Tan X over X
:$\displaystyle \lim_{x \mathop \to 0} \frac {\tan x} x = 1$
Logarithm of Reciprocal
:$\map {\log_b} {\dfrac 1 x} = -\log_b x$
Primitive of Hyperbolic Secant Function/Arcsine Form
:$\ds \int \sech x \rd x = \map \arcsin {\tanh x} + C$
Primitive of Hyperbolic Secant Function/Arctangent of Exponential Form
:$\ds \int \sech x \rd x = 2 \map \arctan {e^x} + C$
Primitive of Hyperbolic Cosecant Function/Hyperbolic Tangent Form
:$\ds \int \csch x \rd x = \ln \size {\tanh \frac x 2} + C$ where $\tanh \dfrac x 2 \ne 0$.
Primitive of Square of Hyperbolic Secant Function
:$\ds \int \sech^2 x \rd x = \tanh x + C$ where $C$ is an arbitrary constant.
Primitive of Square of Hyperbolic Cosecant Function
:$\ds \int \csch^2 x \rd x = -\coth x + C$ where $C$ is an arbitrary constant.
Primitive of Square of Hyperbolic Tangent Function
:$\displaystyle \int \tanh^2 x \rd x = x - \tanh x + C$ where $C$ is an arbitrary constant.
Primitive of Square of Hyperbolic Cotangent Function
:$\displaystyle \int \coth^2 x \rd x = x - \coth x + C$ where $C$ is an arbitrary constant.
Heaviside Step Function is Piecewise Continuous
Let $c \ge 0$ be a constant real number. The Heaviside step function: :$\mu_c \left({t}\right) = \begin{cases} 1 & : t > c \\ 0 & : t < c \end{cases}$ is piecewise continuous for any interval of the form: :$\left[{c - M \,.\,.\, c + M}\right]$ where $M > 0$ is some arbitrarily large constant.
Euler Formula for Sine Function/Complex Numbers/Proof 1/Lemma 1
The function: :$\dfrac {\sinh x} x$  is increasing for positive real $x$.
Hyperbolic Tangent Less than X
:$\tanh x \le x$ for $x \ge 0$.
Primitive of Square of Hyperbolic Sine Function
:$\displaystyle \int \sinh^2 x \rd x = \frac {\sinh 2 x} 4 - \frac x 2 + C$ where $C$ is an arbitrary constant.
Primitive of Square of Hyperbolic Cosine Function
:$\displaystyle \int \cosh^2 x \rd x = \frac {\sinh 2 x} 4 + \frac x 2 + C$ where $C$ is an arbitrary constant.
Primitive of Product of Hyperbolic Cosecant and Cotangent
:$\ds \int \csch x \coth x \rd x = -\csch x + C$ where $C$ is an arbitrary constant.
Primitive of Reciprocal of x squared minus a squared
thumbright600px$\color {blue} {\dfrac 1 {x^2 - a^2} } \qquad \color {green} {-\dfrac 1 a \tanh^{-1} \dfrac x a} \qquad \color {red} {-\dfrac 1 a \coth^{-1} \dfrac x a}$ Let $a \in \R_{>0}$ be a strictly positive real constant.
Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1
:$\displaystyle \int \frac {\d x} {a^2 - x^2} = \begin {cases} \dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} } + C & : \size x < a\\ & \\  \dfrac 1 {2 a} \map \ln {\dfrac {x + a} {x - a} } + C & : \size x > a \\ & \\ \text {undefined} & : \size x = a \end {cases}$
Primitive of Reciprocal of a squared minus x squared
thumbright600px$\color {blue} {\dfrac 1 {a^2 - x^2} } \qquad \color {red} {\dfrac 1 a \tanh^{-1} \dfrac x a} \qquad \color {green} {\dfrac 1 a \coth^{-1} \dfrac x a}$ Let $a \in \R_{>0}$ be a strictly positive real constant.
Primitive of Reciprocal of Root of x squared plus a squared/Logarithm Form
:$\ds \int \frac {\d x} {\sqrt {x^2 + a^2} } = \map \ln {x + \sqrt {x^2 + a^2} } + C$
Limit at Infinity of Polynomial over Complex Exponential
Let $n \in \N$. Let $\map {P_n} x$ be a real polynomial, of degree $n$. Let $e^z$ be the complex exponential function, where $z = x + i y$. Let $a \in \R_{>0}$. Then: :$\displaystyle \lim_{x \mathop \to +\infty} \frac {\map {P_n} x} {e^{a z} } = 0$
Primitives of Trigonometric Functions
This page gathers together primitives of trigonometric functions. In the below, $C$ is an arbitrary constant throughout.
Primitives of Hyperbolic Functions
This page gathers together primitives of hyperbolic functions. In the below, $C$ is an arbitrary constant throughout.
Derivatives of Trigonometric Functions
This page gathers together derivatives of trigonometric functions.
Derivatives of Inverse Trigonometric Functions
This page gathers together derivatives of inverse trigonometric functions.
Derivatives of Hyperbolic Functions
This page gathers together derivatives of hyperbolic functions. Let $u$ be a differentiable real function of $x$.
Derivatives of Inverse Hyperbolic Functions
This page gathers together derivatives of inverse hyperbolic functions.
Primitive of Reciprocal of Root of x squared minus a squared/Logarithm Form
:$\ds \int \frac {\d x} {\sqrt {x^2 - a^2} } = \ln \size {x + \sqrt {x^2 - a^2} } + C$
Primitive of Reciprocal of x by Root of x squared minus a squared
:$\displaystyle \int \frac {\d x} {x \sqrt {x^2 - a^2} } = \frac 1 a \arcsec \size {\frac x a} + C$ for $\size x > a$.
Primitive of Reciprocal of x by Root of x squared plus a squared/Logarithm Form
:$\displaystyle \int \frac {\d x} {x \sqrt {x^2 + a^2} } = -\frac 1 a \map \ln {\frac {a + \sqrt {x^2 + a^2} } x} + C$
Primitive of Reciprocal of x by Root of a squared minus x squared/Logarithm Form
:$\displaystyle \int \frac {\d x} {x \sqrt {a^2 - x^2} } = -\frac 1 a \map \ln {\frac {a + \sqrt {a^2 - x^2} } x} + C$
Generalized Integration by Parts
Let $\map f x, \map g x$ be real functions which are integrable and at least $n$ times differentiable. Then: {{begin-eqn}} {{eqn | l = \int f^{\paren n} g \rd x       | r = \sum_{j \mathop = 0}^{n - 1} \paren {-1}^j f^{\paren {n - j - 1} } g^{\paren j} + \paren {-1}^n \int f g^{\paren n} \rd x       | c =  }} {{eqn | r = f^{\paren {n - 1} } g - f^{\paren {n - 2} } g' + f^{\paren {n - 3} } g'' - \cdots + \paren {-1}^n \int f g^{\paren n} \rd x       | c =  }} {{end-eqn}} where $f^{\paren n}$ denotes the $n$th derivative of $f$.
Primitive of Function of a x + b
:$\displaystyle \int \map F {a x + b} \rd x = \frac 1 a \int \map F u \rd u$ where $u = a x + b$.
Primitive of Function of Root of a x + b
:$\displaystyle \int F \left({\sqrt {a x + b}}\right) \ \mathrm d x = \frac 2 a \int u \ F \left({u}\right) \ \mathrm d u$ where $u = \sqrt {a x + b}$.
Derivative of Nth Root
Let $n \in \N_{>0}$. Let $f: \R \to \R$ be the real function defined as $f \left({x}\right) = \sqrt [n] x$. Then: :$f' \left({x}\right) = \dfrac 1 {n \left({\sqrt [n] x}\right)^{n-1} }$ everywhere that $f \left({x}\right) = \sqrt [n] x$ is defined.
Primitive of Function of Root of a squared minus x squared
:$\displaystyle \int F \left({\sqrt {a^2 - x^2}}\right) \ \mathrm d x = a \int \cos u \ F \left({a \cos u}\right) \ \mathrm d u$ where $x = a \sin u$.
Primitive of Function of Root of x squared minus a squared
:$\displaystyle \int F \left({\sqrt {x^2 - a^2}}\right) \ \mathrm d x = a \int \sec u \tan u \ F \left({a \tan u}\right) \ \mathrm d u$ where $x = a \sec u$.
Primitive of Function of Exponential Function
:$\displaystyle \int \map F {e^{a x} } \rd x = \frac 1 a \int \frac {\map F u} u \rd u$ where $u = e^{a x}$.
Primitive of Function of Arcsine
:$\displaystyle \int F \left({\arcsin \frac x a}\right) \ \mathrm d x = a \int F \left({u}\right) \cos u \ \mathrm d u$ where $u = \arcsin \dfrac x a$.
Primitive of Function of Arccosine
:$\displaystyle \int F \paren {\arccos \frac x a} \rd x = -a \int F \paren u \sin u \rd u$ where $u = \arccos \dfrac x a$.
Primitive of Function of Arctangent
:$\displaystyle \int F \left({\arctan \frac x a}\right) \ \mathrm d x = a \int F \left({u}\right) \sec^2 u \ \mathrm d u$ where $u = \arctan \dfrac x a$.
Primitive of Function of Arccotangent
:$\displaystyle \int F \left({\operatorname{arccot} \frac x a}\right) \ \mathrm d x = -a \int F \left({u}\right) \csc^2 u \ \mathrm d u$ where $u = \operatorname{arccot} \dfrac x a$.
Primitive of Function of Arcsecant
:$\displaystyle \int \map F {\arcsec \frac x a} \rd x = a \int \map F u \sec u \tan u \rd u$ where $u = \arcsec \dfrac x a$.
Primitive of Reciprocal of a x + b
:$\displaystyle \int \frac {\d x} {a x + b} = \frac 1 a \ln \size {a x + b} + C$
Primitive of x over a x + b
:$\displaystyle \int \frac {x \rd x} {a x + b} = \frac x a - \frac b {a^2} \ln \size {a x + b} + C$
Laplace Transform of Exponential times Function
:$\laptrans {e^{a t} \map f t} = \map F {s - a}$ everywhere that $\laptrans f$ exists, for $\map \Re s > a$
Laplace Transform of Derivative
Let $f'$ be piecewise continuous with one-sided limits on said intervals. Then $\laptrans f$ exists for $\map \Re s > a$, and: :$\laptrans {\map {f'} t} = s \laptrans {\map f t} - \map f 0$
Primitive of x squared over a x + b
:$\displaystyle \int \frac {x^2 \rd x} {a x + b} = \frac {\paren {a x + b}^2} {2 a^3} - \frac {2 b \paren {a x + b} } {a^3} + \frac {b^2} {a^3} \ln \size {a x + b} + C$
Primitive of x cubed over a x + b
:$\displaystyle \int \frac {x^3 \rd x} {a x + b} = \frac {\paren {a x + b}^3} {3 a^4} - \frac {3 b \paren {a x + b}^2} {2 a^4} - \frac {3 b^2 \paren {a x + b} } {a^4} + \frac {b^3} {a^4} \ln \size {a x + b} + C$
Primitive of Reciprocal of x by a x + b
:$\displaystyle \int \frac {\rd x} {x \paren {a x + b} } = \frac 1 b \ln \size {\frac x {a x + b} } + C$
Primitive of Reciprocal of x squared by a x + b
:$\ds \int \frac {\d x} {x^2 \paren {a x + b} } = -\frac 1 {b x} + \frac a {b^2} \ln \size {\frac {a x + b} x} + C$
Primitive of Reciprocal of x cubed by a x + b
:$\displaystyle \int \frac {\mathrm d x} {x^3 \left({a x + b}\right)} = \frac {2 a x - b} {2 b^2 x^2} + \frac {a^2} {b^3} \ln \left\vert{\frac x {a x + b} }\right\vert + C$
Primitives of Rational Functions involving a x + b
This page gathers together the primitives of some rational functions involving $a x + b$.
Primitive of Reciprocal of a x + b squared
:$\displaystyle \int \frac {\d x} {\paren {a x + b}^2} = -\frac 1 {a \paren {a x + b} } + C$
Primitive of x over a x + b squared
:$\displaystyle \int \frac {x \ \mathrm d x} {\left({a x + b}\right)^2} = \frac b {a^2 \left({a x + b}\right)} + \frac 1 {a^2} \ln \left\vert{a x + b}\right\vert + C$
Primitive of x squared over a x + b squared
:$\displaystyle \int \frac {x^2 \rd x} {\paren {a x + b}^2} = \frac {a x + b} {a^3} - \frac {b^2} {a^3 \paren {a x + b} } - \frac {2 b} {a^3} \ln \size {a x + b} + C$
Primitive of x cubed over a x + b squared
:$\displaystyle \int \frac {x^3 \ \mathrm d x} {\left({a x + b}\right)^2} = \frac {\left({a x + b}\right)^2} {2 a^4} - \frac {3 b \left({a x + b}\right)} {a^4} + \frac {b^3} {a^4 \left({a x + b}\right)} + \frac {3 b^2} {a^4} \ln \left\vert{a x + b}\right\vert + C$
Primitive of Reciprocal of x by a x + b squared
:$\displaystyle \int \frac {\d x} {x \paren {a x + b}^2} = \frac 1 {b \paren {a x + b} } + \frac 1 {b^2} \ln \size {\frac x {a x + b} } + C$
Primitive of Reciprocal of x squared by a x + b squared
:$\displaystyle \int \frac {\mathrm d x} {x^2 \left({a x + b}\right)^2} = \frac {-a} {b^2 \left({a x + b}\right)} - \frac 1 {b^2 x} + \frac {2 a} {b^3} \ln \left\vert{\frac {a x + b} x}\right\vert + C$
Primitive of Reciprocal of x cubed by a x + b squared
:$\displaystyle \int \frac {\mathrm d x} {x^3 \left({a x + b}\right)^2} = - \frac {\left({a x + b}\right)^2} {2 b^4 x^2} + \frac {3 a \left({a x + b}\right)} {b^4 x} - \frac {a^3 x} {b^4 \left({a x + b}\right)} + \frac {3 a^2} {b^4} \ln \left\vert{\frac x {a x + b} }\right\vert + C$
Primitives of Rational Functions involving a x + b squared
This page gathers together the primitives of some rational functions involving $\left({a x + b}\right)^2$.
Primitive of Reciprocal of a x + b cubed
:$\displaystyle \int \frac {\d x} {\paren {a x + b}^3} = -\frac 1 {2 a \paren {a x + b}^2} + C$