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Equivalent infinitesimal

When two variables $\alpha$ and $\beta$ converge to zero at the same limit point and $\textstyle \lim {\frac {\beta }{\alpha }}=1$, they are called equivalent infinitesimal (equiv. $\alpha \sim \beta$).

Moreover, if variables $\alpha '$ and $\beta '$ are such that $\alpha \sim \alpha '$ and $\beta \sim \beta '$, then:

$\lim {\frac {\beta }{\alpha }}=\lim {\frac {\beta '}{\alpha '}}$

Here is a brief proof:

Suppose there are two equivalent infinitesimals $\alpha \sim \alpha '$ and $\beta \sim \beta '$.

$$\lim {\frac {\beta }{\alpha }}=\lim {\frac {\beta \beta '\alpha '}{\beta '\alpha '\alpha }}=\lim {\frac {\beta }{\beta '}}\lim {\frac {\alpha '}{\alpha }}\lim {\frac {\beta '}{\alpha '}}=\lim {\frac {\beta '}{\alpha '}}$$

For the evaluation of the indeterminate form $0/0$, one can make use of the following facts about equivalent infinitesimals (e.g., $x\sim \sin x$ if x becomes closer to zero):

$x\sim \sin x,$

$x\sim \arcsin x,$

$x\sim \sinh x,$

$x\sim \tan x,$

$x\sim \arctan x,$

$x\sim \ln(1+x),$

$1-\cos x\sim {\frac {x^{2}}{2}},$

$\cosh x-1\sim {\frac {x^{2}}{2}},$

$a^{x}-1\sim x\ln a,$

$e^{x}-1\sim x,$

$(1+x)^{a}-1\sim ax.$

For example:

$${\begin{aligned}\lim _{x\to 0}{\frac {1}{x^{3}}}\left[\left({\frac {2+\cos x}{3}}\right)^{x}-1\right]&=\lim _{x\to 0}{\frac {e^{x\ln {\frac {2+\cos x}{3}}}-1}{x^{3}}}\\&=\lim _{x\to 0}{\frac {1}{x^{2}}}\ln {\frac {2+\cos x}{3}}\\&=\lim _{x\to 0}{\frac {1}{x^{2}}}\ln \left({\frac {\cos x-1}{3}}+1\right)\\&=\lim _{x\to 0}{\frac {\cos x-1}{3x^{2}}}\\&=\lim _{x\to 0}-{\frac {x^{2}}{6x^{2}}}\\&=-{\frac {1}{6}}\end{aligned}}$$

In the 2nd equality, $e^{y}-1\sim y$ where $y=x\ln {2+\cos x \over 3}$ as y become closer to 0 is used, and $y\sim \ln {(1+y)}$ where $y={{\cos x-1} \over 3}$ is used in the 4th equality, and $1-\cos x\sim {x^{2} \over 2}$ is used in the 5th equality.

Differentiation rules

Unless otherwise stated, all functions are functions of real numbers (R) that return real values; although more generally, the formulae below apply wherever they are well defined — including the case of complex numbers (C).

Constant term rule

For any value of $c$, where $c\in \mathbb {R}$, if $f(x)$ is the constant function given by $f(x)=c$, then ${\frac {df}{dx}}=0$.

Proof

Let $c\in \mathbb {R}$ and $f(x)=c$. By the definition of the derivative,

${\begin{aligned}f'(x)&=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}\\&=\lim _{h\to 0}{\frac {(c)-(c)}{h}}\\&=\lim _{h\to 0}{\frac {0}{h}}\\&=\lim _{h\to 0}0\\&=0\end{aligned}}$

This shows that the derivative of any constant function is 0.

Intuitive (geometric) explanation

The derivative of the function at a point is the slope of the line tangent to the curve at the point. Slope of the constant function is zero, because the tangent line to the constant function is horizontal and it's angle is zero.

In other words, the value of the constant function, y, will not change as the value of x increases or decreases.

At each point, the derivative is the slope of a line that is tangent to the curve at that point. Note: the derivative at point A is positive where green and dash–dot, negative where red and dashed, and zero where black and solid.

Differentiation is linear

Main article: Linearity of differentiation

For any functions $f$ and $g$ and any real numbers $a$ and $b$, the derivative of the function $h(x)=af(x)+bg(x)$ with respect to $x$ is: $h'(x)=af'(x)+bg'(x).$

In Leibniz's notation this is written as:

$${\frac {d(af+bg)}{dx}}=a{\frac {df}{dx}}+b{\frac {dg}{dx}}.$$

Special cases include:

• The constant factor rule
  $$(af)'=af'$$
• The sum rule
  $$(f+g)'=f'+g'$$
• The difference rule
  $$(f-g)'=f'-g'.$$

The product rule

Main article: Product rule

For the functions $f$ and $g$, the derivative of the function $h(x)=f(x)g(x)$ with respect to $x$ is

$$h'(x)=(fg)'(x)=f'(x)g(x)+f(x)g'(x).$$

In Leibniz's notation this is written

$${\frac {d(fg)}{dx}}=g{\frac {df}{dx}}+f{\frac {dg}{dx}}.$$

The chain rule

Main article: Chain rule

The derivative of the function $h(x)=f(g(x))$ is

$$h'(x)=f'(g(x))\cdot g'(x).$$

In Leibniz's notation, this is written as:

$${\frac {d}{dx}}h(x)=\left.{\frac {d}{dz}}f(z)\right|_{z=g(x)}\cdot {\frac {d}{dx}}g(x),$$

often abridged to

$${\frac {dh(x)}{dx}}={\frac {df(g(x))}{dg(x)}}\cdot {\frac {dg(x)}{dx}}.$$

Focusing on the notion of maps, and the differential being a map ${\text{D}}$, this is written in a more concise way as:

$$[{\text{D}}(f\circ g)]_{x}=[{\text{D}}f]_{g(x)}\cdot [{\text{D}}g]_{x}\,.$$

The inverse function rule

Main article: Inverse function rule

If the function f has an inverse function g, meaning that $g(f(x))=x$ and $f(g(y))=y,$ then

$$g'={\frac {1}{f'\circ g}}.$$

In Leibniz notation, this is written as

$${\frac {dx}{dy}}={\frac {1}{\frac {dy}{dx}}}.$$

The polynomial or elementary power rule

Main article: Power rule

If $f(x)=x^{r}$, for any real number $r\neq 0,$ then

$f'(x)=rx^{r-1}.$

When $r=1,$ this becomes the special case that if $f(x)=x,$ then $f'(x)=1.$

Combining the power rule with the sum and constant multiple rules permits the computation of the derivative of any polynomial.

The reciprocal rule

Main article: Reciprocal rule

The derivative of $h(x)={\frac {1}{f(x)}}$for any (nonvanishing) function f is:

$h'(x)=-{\frac {f'(x)}{(f(x))^{2}}}$ wherever f is non-zero.

In Leibniz's notation, this is written

${\frac {d(1/f)}{dx}}=-{\frac {1}{f^{2}}}{\frac {df}{dx}}.$

The reciprocal rule can be derived either from the quotient rule, or from the combination of power rule and chain rule.

The quotient rule

Main article: Quotient rule

If f and g are functions, then:

$\left({\frac {f}{g}}\right)'={\frac {f'g-g'f}{g^{2}}}\quad$ wherever g is nonzero.

This can be derived from the product rule and the reciprocal rule.

Generalized power rule

Main article: Power rule

The elementary power rule generalizes considerably. The most general power rule is the functional power rule: for any functions f and g,

$(f^{g})'=\left(e^{g\ln f}\right)'=f^{g}\left(f'{g \over f}+g'\ln f\right),\quad$

wherever both sides are well defined.

Special cases

• If ${\textstyle f(x)=x^{a}\!}$, then ${\textstyle f'(x)=ax^{a-1}}$ when a is any non-zero real number and x is positive.
• The reciprocal rule may be derived as the special case where ${\textstyle g(x)=-1\!}$.

${\frac {d}{dx}}\left(c^{ax}\right)={ac^{ax}\ln c},\qquad c>0$

the equation above is true for all c, but the derivative for ${\textstyle c<0}$ yields a complex number.

${\frac {d}{dx}}\left(e^{ax}\right)=ae^{ax}$

${\frac {d}{dx}}\left(\log _{c}x\right)={1 \over x\ln c},\qquad c>1$

the equation above is also true for all c, but yields a complex number if ${\textstyle c<0\!}$.

${\frac {d}{dx}}\left(\ln x\right)={1 \over x},\qquad x>0.$

${\frac {d}{dx}}\left(\ln |x|\right)={1 \over x},\qquad x\neq 0.$

${\frac {d}{dx}}\left(W(x)\right)={1 \over {x+e^{W(x)}}},\qquad x>-{1 \over e}.\qquad$where $W(x)$ is the Lambert W function

${\frac {d}{dx}}\left(x^{x}\right)=x^{x}(1+\ln x).$

${\frac {d}{dx}}\left(f(x)^{g(x)}\right)=g(x)f(x)^{g(x)-1}{\frac {df}{dx}}+f(x)^{g(x)}\ln {(f(x))}{\frac {dg}{dx}},\qquad {\text{if }}f(x)>0,{\text{ and if }}{\frac {df}{dx}}{\text{ and }}{\frac {dg}{dx}}{\text{ exist.}}$

${\frac {d}{dx}}\left(f_{1}(x)^{f_{2}(x)^{\left(...\right)^{f_{n}(x)}}}\right)=\left[\sum \limits _{k=1}^{n}{\frac {\partial }{\partial x_{k}}}\left(f_{1}(x_{1})^{f_{2}(x_{2})^{\left(...\right)^{f_{n}(x_{n})}}}\right)\right]{\biggr \vert }_{x_{1}=x_{2}=...=x_{n}=x},{\text{ if }}f_{i<n}(x)>0{\text{ and }}$ ${\frac {df_{i}}{dx}}{\text{ exists. }}$

Logarithmic derivatives

The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule):

$(\ln f)'={\frac {f'}{f}}\quad$ wherever f is positive.

Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative.

Logarithms can be used to remove exponents, convert products into sums, and convert division into subtraction — each of which may lead to a simplified expression for taking derivatives.

Main article: Differentiation of trigonometric functions

${\frac {d}{dx}}\sin x=\cos x$	${\frac {d}{dx}}\arcsin x={\frac {1}{\sqrt {1-x^{2}}}}$
${\frac {d}{dx}}\cos x=-\sin x$	${\frac {d}{dx}}\arccos x=-{\frac {1}{\sqrt {1-x^{2}}}}$
${\frac {d}{dx}}\tan x=\sec ^{2}x={\frac {1}{\cos ^{2}x}}=1+\tan ^{2}x$	${\frac {d}{dx}}\arctan x={\frac {1}{1+x^{2}}}$
${\frac {d}{dx}}\csc x=-\csc {x}\cot {x}$	${\frac {d}{dx}}\operatorname {arccsc} x=-{\frac {1}{|x|{\sqrt {x^{2}-1}}}}$
${\frac {d}{dx}}\sec x=\sec {x}\tan {x}$	${\frac {d}{dx}}\operatorname {arcsec} x={\frac {1}{|x|{\sqrt {x^{2}-1}}}}$
${\frac {d}{dx}}\cot x=-\csc ^{2}x=-{\frac {1}{\sin ^{2}x}}=-1-\cot ^{2}x$	${\frac {d}{dx}}\operatorname {arccot} x=-{1 \over 1+x^{2}}$

The derivatives in the table above are for when the range of the inverse secant is $[0,\pi ]\!$ and when the range of the inverse cosecant is $\left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right].$

It is common to additionally define an inverse tangent function with two arguments, $\arctan(y,x).$ Its value lies in the range $[-\pi ,\pi ]$ and reflects the quadrant of the point $(x,y).$ For the first and fourth quadrant (i.e. $x>0$) one has $\arctan(y,x>0)=\arctan(y/x).$ Its partial derivatives are

${\frac {\partial \arctan(y,x)}{\partial y}}={\frac {x}{x^{2}+y^{2}}}\qquad {\text{and}}\qquad {\frac {\partial \arctan(y,x)}{\partial x}}={\frac {-y}{x^{2}+y^{2}}}.$

Maclaurin Series

See also: List of mathematical series

Several important Maclaurin series expansions follow. All these expansions are valid for complex arguments x.

Exponential function

The exponential function e^x (in blue), and the sum of the first n + 1 terms of its Taylor series at 0 (in red).

The exponential function $e^{x}$ (with base e) has Maclaurin series

$$e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+\cdots .$$

It converges for all x.

The exponential generating function of the Bell numbers is the exponential function of the predecessor of the exponential function:

$$\exp(\exp {x}-1)=\sum _{n=0}^{\infty }{\frac {B_{n}}{n!}}x^{n}$$

Natural logarithm

The natural logarithm (with base e) has Maclaurin series

$${\begin{aligned}\ln(1-x)&=-\sum _{n=1}^{\infty }{\frac {x^{n}}{n}}=-x-{\frac {x^{2}}{2}}-{\frac {x^{3}}{3}}-\cdots ,\\\ln(1+x)&=\sum _{n=1}^{\infty }(-1)^{n+1}{\frac {x^{n}}{n}}=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-\cdots .\end{aligned}}$$

The last series is known as Mercator series, named after Nicholas Mercator (since it was published in his 1668 treatise Logarithmotechnia). Both of these series converge for $|x|<1$. (In addition, the series for ln(1 − x) converges for x = −1, and the series for ln(1 + x) converges for x = 1.)

Geometric series

The geometric series and its derivatives have Maclaurin series

$${\begin{aligned}{\frac {1}{1-x}}&=\sum _{n=0}^{\infty }x^{n}\\{\frac {1}{(1-x)^{2}}}&=\sum _{n=1}^{\infty }nx^{n-1}\\{\frac {1}{(1-x)^{3}}}&=\sum _{n=2}^{\infty }{\frac {(n-1)n}{2}}x^{n-2}.\end{aligned}}$$

All are convergent for $|x|<1$. These are special cases of the binomial series given in the next section.

Binomial series

The binomial series is the power series

$$(1+x)^{\alpha }=\sum _{n=0}^{\infty }{\binom {\alpha }{n}}x^{n}$$

whose coefficients are the generalized binomial coefficients

$${\binom {\alpha }{n}}=\prod _{k=1}^{n}{\frac {\alpha -k+1}{k}}={\frac {\alpha (\alpha -1)\cdots (\alpha -n+1)}{n!}}.$$

(If n = 0, this product is an empty product and has value 1.) It converges for $|x|<1$ for any real or complex number α.

When α = −1, this is essentially the infinite geometric series mentioned in the previous section. The special cases α = 1/2 and α = −1/2 give the square root function and its inverse:

$${\begin{aligned}(1+x)^{\frac {1}{2}}&=1+{\frac {1}{2}}x-{\frac {1}{8}}x^{2}+{\frac {1}{16}}x^{3}-{\frac {5}{128}}x^{4}+{\frac {7}{256}}x^{5}-\cdots &=\sum _{n=0}^{\infty }{\frac {(-1)^{n-1}(2n)!}{4^{n}(n!)^{2}(2n-1)}}x^{n},\\(1+x)^{-{\frac {1}{2}}}&=1-{\frac {1}{2}}x+{\frac {3}{8}}x^{2}-{\frac {5}{16}}x^{3}+{\frac {35}{128}}x^{4}-{\frac {63}{256}}x^{5}+\cdots &=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n)!}{4^{n}(n!)^{2}}}x^{n}.\end{aligned}}$$

When only the linear term is retained, this simplifies to the binomial approximation.

Trigonometric functions

The usual trigonometric functions and their inverses have the following Maclaurin series:

$${\begin{aligned}\sin x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}&&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-\cdots &&{\text{for all }}x\\[6pt]\cos x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}&&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots &&{\text{for all }}x\\[6pt]\tan x&=\sum _{n=1}^{\infty }{\frac {B_{2n}(-4)^{n}\left(1-4^{n}\right)}{(2n)!}}x^{2n-1}&&=x+{\frac {x^{3}}{3}}+{\frac {2x^{5}}{15}}+\cdots &&{\text{for }}|x|<{\frac {\pi }{2}}\\[6pt]\sec x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}E_{2n}}{(2n)!}}x^{2n}&&=1+{\frac {x^{2}}{2}}+{\frac {5x^{4}}{24}}+\cdots &&{\text{for }}|x|<{\frac {\pi }{2}}\\[6pt]\arcsin x&=\sum _{n=0}^{\infty }{\frac {(2n)!}{4^{n}(n!)^{2}(2n+1)}}x^{2n+1}&&=x+{\frac {x^{3}}{6}}+{\frac {3x^{5}}{40}}+\cdots &&{\text{for }}|x|\leq 1\\[6pt]\arccos x&={\frac {\pi }{2}}-\arcsin x\\&={\frac {\pi }{2}}-\sum _{n=0}^{\infty }{\frac {(2n)!}{4^{n}(n!)^{2}(2n+1)}}x^{2n+1}&&={\frac {\pi }{2}}-x-{\frac {x^{3}}{6}}-{\frac {3x^{5}}{40}}-\cdots &&{\text{for }}|x|\leq 1\\[6pt]\arctan x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}x^{2n+1}&&=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-\cdots &&{\text{for }}|x|\leq 1,\ x\neq \pm i\end{aligned}}$$

All angles are expressed in radians. The numbers B_k appearing in the expansions of tan x are the Bernoulli numbers. The E_k in the expansion of sec x are Euler numbers.

Hyperbolic functions

The hyperbolic functions have Maclaurin series closely related to the series for the corresponding trigonometric functions:

$${\begin{aligned}\sinh x&=\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{(2n+1)!}}&&=x+{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}+\cdots &&{\text{for all }}x\\[6pt]\cosh x&=\sum _{n=0}^{\infty }{\frac {x^{2n}}{(2n)!}}&&=1+{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}+\cdots &&{\text{for all }}x\\[6pt]\tanh x&=\sum _{n=1}^{\infty }{\frac {B_{2n}4^{n}\left(4^{n}-1\right)}{(2n)!}}x^{2n-1}&&=x-{\frac {x^{3}}{3}}+{\frac {2x^{5}}{15}}-{\frac {17x^{7}}{315}}+\cdots &&{\text{for }}|x|<{\frac {\pi }{2}}\\[6pt]\operatorname {arsinh} x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n)!}{4^{n}(n!)^{2}(2n+1)}}x^{2n+1}&&=x-{\frac {x^{3}}{6}}+{\frac {3x^{5}}{40}}-\cdots &&{\text{for }}|x|\leq 1\\[6pt]\operatorname {artanh} x&=\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{2n+1}}&&=x+{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}+\cdots &&{\text{for }}|x|\leq 1,\ x\neq \pm 1\end{aligned}}$$

The numbers B_k appearing in the series for tanh x are the Bernoulli numbers.

Polylogarithmic functions

The polylogarithms have these defining identities:

$${\begin{aligned}{\text{Li}}_{2}(x)&=\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}x^{n}\\{\text{Li}}_{3}(x)&=\sum _{n=1}^{\infty }{\frac {1}{n^{3}}}x^{n}\end{aligned}}$$

The Legendre chi functions are defined as follows:

$${\begin{aligned}\chi _{2}(x)&=\sum _{n=0}^{\infty }{\frac {1}{(2n+1)^{2}}}x^{2n+1}\\\chi _{3}(x)&=\sum _{n=0}^{\infty }{\frac {1}{(2n+1)^{3}}}x^{2n+1}\end{aligned}}$$

And the formulas presented below are called inverse tangent integrals:

$${\begin{aligned}{\text{Ti}}_{2}(x)&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{2}}}x^{2n+1}\\{\text{Ti}}_{3}(x)&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{3}}}x^{2n+1}\end{aligned}}$$

In statistical thermodynamics these formulas are of great importance.

Elliptic functions

The complete elliptic integrals of first kind K and of second kind E can be defined as follows:

$${\begin{aligned}{\frac {2}{\pi }}K(x)&=\sum _{n=0}^{\infty }{\frac {[(2n)!]^{2}}{16^{n}(n!)^{4}}}x^{2n}\\{\frac {2}{\pi }}E(x)&=\sum _{n=0}^{\infty }{\frac {[(2n)!]^{2}}{(1-2n)16^{n}(n!)^{4}}}x^{2n}\end{aligned}}$$

The Jacobi theta functions describe the world of the elliptic modular functions and they have these Taylor series:

$${\begin{aligned}\vartheta _{00}(x)&=1+2\sum _{n=1}^{\infty }x^{n^{2}}\\\vartheta _{01}(x)&=1+2\sum _{n=1}^{\infty }(-1)^{n}x^{n^{2}}\end{aligned}}$$

The regular partition number sequence P(n) has this generating function:

$$\vartheta _{00}(x)^{-1/6}\vartheta _{01}(x)^{-2/3}{\biggl [}{\frac {\vartheta _{00}(x)^{4}-\vartheta _{01}(x)^{4}}{16\,x}}{\biggr ]}^{-1/24}=\sum _{n=0}^{\infty }P(n)x^{n}=\prod _{k=1}^{\infty }{\frac {1}{1-x^{k}}}$$

The strict partition number sequence Q(n) has that generating function:

$$\vartheta _{00}(x)^{1/6}\vartheta _{01}(x)^{-1/3}{\biggl [}{\frac {\vartheta _{00}(x)^{4}-\vartheta _{01}(x)^{4}}{16\,x}}{\biggr ]}^{1/24}=\sum _{n=0}^{\infty }Q(n)x^{n}=\prod _{k=1}^{\infty }{\frac {1}{1-x^{2k-1}}}$$

Integration rules

C is used for an arbitrary constant of integration that can only be determined if something about the value of the integral at some point is known. Thus, each function has an infinite number of antiderivatives.

These formulas only state in another form the assertions in the table of derivatives.

Integrals with a singularity

When there is a singularity in the function being integrated such that the antiderivative becomes undefined or at some point (the singularity), then C does not need to be the same on both sides of the singularity. The forms below normally assume the Cauchy principal value around a singularity in the value of C but this is in general, not necessary. For instance in

$$\int {1 \over x}\,dx=\ln \left|x\right|+C$$

there is a singularity at 0 and the antiderivative becomes infinite there. If the integral above were to be used to compute a definite integral between −1 and 1, one would get the wrong answer 0. This however is the Cauchy principal value of the integral around the singularity. If the integration is done in the complex plane the result depends on the path around the origin, in this case the singularity contributes −iπ when using a path above the origin and iπ for a path below the origin. A function on the real line could use a completely different value of C on either side of the origin as in:

$$\int {1 \over x}\,dx=\ln |x|+{\begin{cases}A&{\text{if }}x>0;\\B&{\text{if }}x<0.\end{cases}}$$

Rational functions

See also: List of integrals of rational functions

• $\int a\,dx=ax+C$

The following function has a non-integrable singularity at 0 for n ≤ −1:

• $\int x^{n}\,dx={\frac {x^{n+1}}{n+1}}+C\qquad {\text{(for }}n\neq -1{\text{)}}$ (Cavalieri's quadrature formula)
• $\int (ax+b)^{n}\,dx={\frac {(ax+b)^{n+1}}{a(n+1)}}+C\qquad {\text{(for }}n\neq -1{\text{)}}$
• $\int {1 \over x}\,dx=\ln \left|x\right|+C$
  • More generally,
    $$\int {1 \over x}\,dx={\begin{cases}\ln \left|x\right|+C^{-}&x<0\\\ln \left|x\right|+C^{+}&x>0\end{cases}}$$
• $\int {\frac {c}{ax+b}}\,dx={\frac {c}{a}}\ln \left|ax+b\right|+C$

Exponential functions

See also: List of integrals of exponential functions

• $\int e^{ax}\,dx={\frac {1}{a}}e^{ax}+C$
• $\int f'(x)e^{f(x)}\,dx=e^{f(x)}+C$
• $\int a^{x}\,dx={\frac {a^{x}}{\ln a}}+C$
• $\int {e^{x}\left(f\left(x\right)+f'\left(x\right)\right)\,dx}=e^{x}f\left(x\right)+C$
• $\int {e^{x}\left(f\left(x\right)-\left(-1\right)^{n}{\frac {d^{n}f\left(x\right)}{dx^{n}}}\right)\,dx}=e^{x}\sum _{k=1}^{n}{\left(-1\right)^{k-1}{\frac {d^{k-1}f\left(x\right)}{dx^{k-1}}}}+C$
  (if $n$ is a positive integer)
• $\int {e^{-x}\left(f\left(x\right)-{\frac {d^{n}f\left(x\right)}{dx^{n}}}\right)\,dx}=-e^{-x}\sum _{k=1}^{n}{\frac {d^{k-1}f\left(x\right)}{dx^{k-1}}}+C$
  (if $n$ is a positive integer)

Logarithms

See also: List of integrals of logarithmic functions

• $\int \ln x\,dx=x\ln x-x+C=x(\ln x-1)+C$
• $\int \log _{a}x\,dx=x\log _{a}x-{\frac {x}{\ln a}}+C={\frac {x}{\ln a}}(\ln x-1)+C$

Trigonometric functions

See also: List of integrals of trigonometric functions

• $\int \sin {x}\,dx=-\cos {x}+C$
• $\int \cos {x}\,dx=\sin {x}+C$
• $\int \tan {x}\,dx=\ln {\left|\sec {x}\right|}+C=-\ln {\left|\cos {x}\right|}+C$
• $\int \cot {x}\,dx=-\ln {\left|\csc {x}\right|}+C=\ln {\left|\sin {x}\right|}+C$
• $\int \sec {x}\,dx=\ln {\left|\sec {x}+\tan {x}\right|}+C=\ln \left|\tan \left({\dfrac {x}{2}}+{\dfrac {\pi }{4}}\right)\right|+C$
  • (See Integral of the secant function. This result was a well-known conjecture in the 17th century.)
• $\int \csc {x}\,dx=-\ln {\left|\csc {x}+\cot {x}\right|}+C=\ln {\left|\csc {x}-\cot {x}\right|}+C=\ln {\left|\tan {\frac {x}{2}}\right|}+C$
• $\int \sec ^{2}x\,dx=\tan x+C$
• $\int \csc ^{2}x\,dx=-\cot x+C$
• $\int \sec {x}\,\tan {x}\,dx=\sec {x}+C$
• $\int \csc {x}\,\cot {x}\,dx=-\csc {x}+C$
• $\int \sin ^{2}x\,dx={\frac {1}{2}}\left(x-{\frac {\sin 2x}{2}}\right)+C={\frac {1}{2}}(x-\sin x\cos x)+C$
• $\int \cos ^{2}x\,dx={\frac {1}{2}}\left(x+{\frac {\sin 2x}{2}}\right)+C={\frac {1}{2}}(x+\sin x\cos x)+C$
• $\int \tan ^{2}x\,dx=\tan x-x+C$
• $\int \cot ^{2}x\,dx=-\cot x-x+C$
• $\int \sec ^{3}x\,dx={\frac {1}{2}}(\sec x\tan x+\ln |\sec x+\tan x|)+C$
  • (See integral of secant cubed.)
• $\int \csc ^{3}x\,dx={\frac {1}{2}}(-\csc x\cot x+\ln |\csc x-\cot x|)+C={\frac {1}{2}}\left(\ln \left|\tan {\frac {x}{2}}\right|-\csc x\cot x\right)+C$
• $\int \sin ^{n}x\,dx=-{\frac {\sin ^{n-1}{x}\cos {x}}{n}}+{\frac {n-1}{n}}\int \sin ^{n-2}{x}\,dx$
• $\int \cos ^{n}x\,dx={\frac {\cos ^{n-1}{x}\sin {x}}{n}}+{\frac {n-1}{n}}\int \cos ^{n-2}{x}\,dx$

Inverse trigonometric functions

See also: List of integrals of inverse trigonometric functions

• $\int \arcsin {x}\,dx=x\arcsin {x}+{\sqrt {1-x^{2}}}+C,{\text{ for }}\vert x\vert \leq 1$
• $\int \arccos {x}\,dx=x\arccos {x}-{\sqrt {1-x^{2}}}+C,{\text{ for }}\vert x\vert \leq 1$
• $\int \arctan {x}\,dx=x\arctan {x}-{\frac {1}{2}}\ln {\vert 1+x^{2}\vert }+C,{\text{ for all real }}x$
• $\int \operatorname {arccot} {x}\,dx=x\operatorname {arccot} {x}+{\frac {1}{2}}\ln {\vert 1+x^{2}\vert }+C,{\text{ for all real }}x$
• $\int \operatorname {arcsec} {x}\,dx=x\operatorname {arcsec} {x}-\ln \left\vert x\,\left(1+{\sqrt {1-x^{-2}}}\,\right)\right\vert +C,{\text{ for }}\vert x\vert \geq 1$
• $\int \operatorname {arccsc} {x}\,dx=x\operatorname {arccsc} {x}+\ln \left\vert x\,\left(1+{\sqrt {1-x^{-2}}}\,\right)\right\vert +C,{\text{ for }}\vert x\vert \geq 1$

Hyperbolic functions

See also: List of integrals of hyperbolic functions

• $\int \sinh x\,dx=\cosh x+C$
• $\int \cosh x\,dx=\sinh x+C$
• $\int \tanh x\,dx=\ln \,(\cosh x)+C$
• $\int \coth x\,dx=\ln |\sinh x|+C,{\text{ for }}x\neq 0$
• $\int \operatorname {sech} \,x\,dx=\arctan \,(\sinh x)+C$
• $\int \operatorname {csch} \,x\,dx=\ln |\operatorname {coth} x-\operatorname {csch} x|+C=\ln \left|\tanh {x \over 2}\right|+C,{\text{ for }}x\neq 0$
• $\int \operatorname {sech} ^{2}x\,dx=\tanh x+C$
• $\int \operatorname {csch} ^{2}x\,dx=-\operatorname {coth} x+C$
• $\int \operatorname {sech} {x}\,\operatorname {tanh} {x}\,dx=-\operatorname {sech} {x}+C$
• $\int \operatorname {csch} {x}\,\operatorname {coth} {x}\,dx=-\operatorname {csch} {x}+C$

Inverse hyperbolic functions

See also: List of integrals of inverse hyperbolic functions

• $\int \operatorname {arcsinh} \,x\,dx=x\,\operatorname {arcsinh} \,x-{\sqrt {x^{2}+1}}+C,{\text{ for all real }}x$
• $\int \operatorname {arccosh} \,x\,dx=x\,\operatorname {arccosh} \,x-{\sqrt {x^{2}-1}}+C,{\text{ for }}x\geq 1$
• $\int \operatorname {arctanh} \,x\,dx=x\,\operatorname {arctanh} \,x+{\frac {\ln \left(\,1-x^{2}\right)}{2}}+C,{\text{ for }}\vert x\vert <1$
• $\int \operatorname {arccoth} \,x\,dx=x\,\operatorname {arccoth} \,x+{\frac {\ln \left(x^{2}-1\right)}{2}}+C,{\text{ for }}\vert x\vert >1$
• $\int \operatorname {arcsech} \,x\,dx=x\,\operatorname {arcsech} \,x+\arcsin x+C,{\text{ for }}0<x\leq 1$
• $\int \operatorname {arccsch} \,x\,dx=x\,\operatorname {arccsch} \,x+\vert \operatorname {arcsinh} \,x\vert +C,{\text{ for }}x\neq 0$

Products of functions proportional to their second derivatives

• $\int \cos ax\,e^{bx}\,dx={\frac {e^{bx}}{a^{2}+b^{2}}}\left(a\sin ax+b\cos ax\right)+C$
• $\int \sin ax\,e^{bx}\,dx={\frac {e^{bx}}{a^{2}+b^{2}}}\left(b\sin ax-a\cos ax\right)+C$
• $\int \cos ax\,\cosh bx\,dx={\frac {1}{a^{2}+b^{2}}}\left(a\sin ax\,\cosh bx+b\cos ax\,\sinh bx\right)+C$
• $\int \sin ax\,\cosh bx\,dx={\frac {1}{a^{2}+b^{2}}}\left(b\sin ax\,\sinh bx-a\cos ax\,\cosh bx\right)+C$

Absolute-value functions

Let f be a continuous function, that has at most one zero. If f has a zero, let g be the unique antiderivative of f that is zero at the root of f; otherwise, let g be any antiderivative of f. Then

$$\int \left|f(x)\right|\,dx=\operatorname {sgn}(f(x))g(x)+C,$$

where sgn(x) is the sign function, which takes the values −1, 0, 1 when x is respectively negative, zero or positive.

This can be proved by computing the derivative of the right-hand side of the formula, taking into account that the condition on g is here for insuring the continuity of the integral.

This gives the following formulas (where a ≠ 0), which are valid over any interval where f is continuous (over larger intervals, the constant C must be replaced by a piecewise constant function):

• $\int \left|(ax+b)^{n}\right|\,dx=\operatorname {sgn}(ax+b){(ax+b)^{n+1} \over a(n+1)}+C$
  when n is odd, and $n\neq -1$.
• $\int \left|\tan {ax}\right|\,dx=-{\frac {1}{a}}\operatorname {sgn}(\tan {ax})\ln(\left|\cos {ax}\right|)+C$
  when ${\textstyle ax\in \left(n\pi -{\frac {\pi }{2}},n\pi +{\frac {\pi }{2}}\right)}$ for some integer n.
• $\int \left|\csc {ax}\right|\,dx=-{\frac {1}{a}}\operatorname {sgn}(\csc {ax})\ln(\left|\csc {ax}+\cot {ax}\right|)+C$
  when $ax\in \left(n\pi ,n\pi +\pi \right)$ for some integer n.
• $\int \left|\sec {ax}\right|\,dx={\frac {1}{a}}\operatorname {sgn}(\sec {ax})\ln(\left|\sec {ax}+\tan {ax}\right|)+C$
  when ${\textstyle ax\in \left(n\pi -{\frac {\pi }{2}},n\pi +{\frac {\pi }{2}}\right)}$ for some integer n.
• $\int \left|\cot {ax}\right|\,dx={\frac {1}{a}}\operatorname {sgn}(\cot {ax})\ln(\left|\sin {ax}\right|)+C$
  when $ax\in \left(n\pi ,n\pi +\pi \right)$ for some integer n.

If the function f does not have any continuous antiderivative which takes the value zero at the zeros of f (this is the case for the sine and the cosine functions), then sgn(f(x)) ∫ f(x) dx is an antiderivative of f on every interval on which f is not zero, but may be discontinuous at the points where f(x) = 0. For having a continuous antiderivative, one has thus to add a well chosen step function. If we also use the fact that the absolute values of sine and cosine are periodic with period π, then we get:

• $\int \left|\sin {ax}\right|\,dx={2 \over a}\left\lfloor {\frac {ax}{\pi }}\right\rfloor -{1 \over a}\cos {\left(ax-\left\lfloor {\frac {ax}{\pi }}\right\rfloor \pi \right)}+C$
• $\int \left|\cos {ax}\right|\,dx={2 \over a}\left\lfloor {\frac {ax}{\pi }}+{\frac {1}{2}}\right\rfloor +{1 \over a}\sin {\left(ax-\left\lfloor {\frac {ax}{\pi }}+{\frac {1}{2}}\right\rfloor \pi \right)}+C$

Special functions

Ci, Si: Trigonometric integrals, Ei: Exponential integral, li: Logarithmic integral function, erf: Error function

• $\int \operatorname {Ci} (x)\,dx=x\operatorname {Ci} (x)-\sin x$
• $\int \operatorname {Si} (x)\,dx=x\operatorname {Si} (x)+\cos x$
• $\int \operatorname {Ei} (x)\,dx=x\operatorname {Ei} (x)-e^{x}$
• $\int \operatorname {li} (x)\,dx=x\operatorname {li} (x)-\operatorname {Ei} (2\ln x)$
• $\int {\frac {\operatorname {li} (x)}{x}}\,dx=\ln x\,\operatorname {li} (x)-x$
• $\int \operatorname {erf} (x)\,dx={\frac {e^{-x^{2}}}{\sqrt {\pi }}}+x\operatorname {erf} (x)$

Tangent Half-angle Substitution

The tangent half-angle substitution relates an angle to the slope of a line.

Introducing a new variable ${\textstyle t=\tan {\tfrac {x}{2}},}$ sines and cosines can be expressed as rational functions of $t,$ and $dx$ can be expressed as the product of $dt$ and a rational function of $t,$ as follows:

$$\sin x={\frac {2t}{1+t^{2}}},\quad \cos x={\frac {1-t^{2}}{1+t^{2}}},\quad {\text{and}}\quad dx={\frac {2}{1+t^{2}}}\,dt.$$

Similar expressions can be written for tan x, cot x, sec x, and csc x.

Derivation

Using the double-angle formulas $\sin x=2\sin {\tfrac {x}{2}}\cos {\tfrac {x}{2}}$ and $\cos x=\cos ^{2}{\tfrac {x}{2}}-\sin ^{2}{\tfrac {x}{2}}$ and introducing denominators equal to one by the Pythagorean identity $1=\cos ^{2}{\tfrac {x}{2}}+\sin ^{2}{\tfrac {x}{2}}$ results in

$${\begin{aligned}\sin x&={\frac {2\sin {\tfrac {x}{2}}\,\cos {\tfrac {x}{2}}}{\cos ^{2}{\tfrac {x}{2}}+\sin ^{2}{\tfrac {x}{2}}}}={\frac {2\tan {\tfrac {x}{2}}}{1+\tan ^{2}{\tfrac {x}{2}}}}={\frac {2t}{1+t^{2}}},\\[18mu]\cos x&={\frac {\cos ^{2}{\tfrac {x}{2}}-\sin ^{2}{\tfrac {x}{2}}}{\cos ^{2}{\tfrac {x}{2}}+\sin ^{2}{\tfrac {x}{2}}}}={\frac {1-\tan ^{2}{\tfrac {x}{2}}}{1+\tan ^{2}{\tfrac {x}{2}}}}={\frac {1-t^{2}}{1+t^{2}}}.\end{aligned}}$$

Finally, since ${\textstyle t=\tan {\tfrac {x}{2}}}$, differentiation rules imply

$$dt={\tfrac {1}{2}}\left(1+\tan ^{2}{\tfrac {x}{2}}\right)dx={\frac {1+t^{2}}{2}}\,dx,$$

and thus

$$dx={\frac {2}{1+t^{2}}}\,dt.$$