int

Definite and indefinite integrals

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Syntax

F = int(expr)

F = int(expr,var)

F = int(expr,a,b)

F = int(expr,var,a,b)

F = int(___,Name,Value)

Description

example

F= int(expr)computes the indefinite integral ofexpr.intuses the default integration variable determined bysymvar(expr,1). Ifexpris a constant, then the default integration variable isx.

example

F= int(expr,var)computes the indefinite integral ofexprwith respect to the symbolic scalar variablevar.

example

F= int(expr,a,b)computes the definite integral ofexprfromatob.intuses the default integration variable determined bysymvar(expr,1). Ifexpris a constant, then the default integration variable isx.

int(expr,[a b])相当于int(expr,a,b).

example

F= int(expr,var,a,b)computes the definite integral ofexprwith respect to the symbolic scalar variablevarfromatob.

int(expr,var,[a b])相当于int(expr,var,a,b).

example

F= int(___,Name,Value)specifies additional options using one or moreName,Valuepair arguments. For example,'IgnoreAnalyticConstraints',truespecifies thatintapplies additional simplifications to the integrand.

Examples

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Indefinite Integral of Univariate Expression

Open Live Script

Define a univariate expression.

symsxexpr = -2*x/(1+x^2)^2;

Find the indefinite integral of the univariate expression.

F = int(expr)

F =
                       
                         $\frac{1}{x^{2} + 1}$

Indefinite Integrals of Multivariate Function

Open Live Script

Define a multivariate function with variablesxandz.

symsxzf(x,z) = x/(1+z^2);

Find the indefinite integrals of the multivariate expression with respect to the variablesxandz.

Fx = int(f,x)

Fx(x, z) =
                       
                         $\frac{x^{2}}{2 (z^{2} + 1)}$

Fz = int(f,z)

Fz(x, z) =
                        $x atan (z)$

If you do not specify the integration variable, thenintuses the first variable returned bysymvaras the integration variable.

var = symvar(f,1)

var =
                        $x$

F = int(f)

F(x, z) =
                       
                         $\frac{x^{2}}{2 (z^{2} + 1)}$

Definite Integrals of Symbolic Expressions

Open Live Script

Integrate a symbolic expression from0to1.

symsxexpr = x*log(1+x); F = int(expr,[0 1])

F =
                       
                         $\frac{1}{4}$

Integrate another expression fromsin(t)to1.

symstF = int(2*x,[sin(t) 1])

F =
                        ${\cos (t)}^{2}$

Whenintcannot compute the value of a definite integral, numerically approximate the integral by usingvpa.

symsxf = cos(x)/sqrt(1 + x^2); Fint = int(f,x,[0 10])

Fint =
                       
                         $\int_{0}^{10} \frac{\cos (x)}{\sqrt{x^{2} + 1}} d x$

Fvpa = vpa(Fint)

Fvpa =
                        $0.37570628299079723478493405557162$

To approximate integrals directly, usevpaintegralinstead ofvpa. Thevpaintegralfunction is faster and provides control over integration tolerances.

Fvpaint = vpaintegral(f,x,[0 10])

Fvpaint =
                        $0.375706$

Integrals of Matrix Elements

Open Live Script

Define a symbolic matrix containing four expressions as its elements.

symsaxtzM = [exp(t) exp(a*t); sin(t) cos(t)]

M =
                       
                         $(\begin{array}{c} e^{t} & e^{a t} \\ \sin (t) & \cos (t) \end{array})$

Find indefinite integrals of the matrix element-wise.

F = int(M,t)

F =
                       
                         $(\begin{array}{c} e^{t} & \frac{e^{a t}}{a} \\ - \cos (t) & \sin (t) \end{array})$

Apply IgnoreAnalyticConstraints

Open Live Script

Define a symbolic function and compute its indefinite integral.

symsf(x)f(x) = acos(cos(x)); F = int(f,x)

F(x) =
                       
                         $x acos (\cos (x)) - \frac{x^{2}}{2 sign (\sin (x))}$

By default,intuses strict mathematical rules. These rules do not letintrewriteacos(cos(x))asx.

If you want a simple practical solution, set'IgnoreAnalyticConstraints'totrue.

F = int(f,x,'IgnoreAnalyticConstraints',true)

F(x) =
                       
                         $\frac{x^{2}}{2}$

Ignore Special Cases

Open Live Script

Define a symbolic expression $x^{t}$ and compute its indefinite integral with respect to the variable $x$ .

symsxtF = int(x^t,x)

F =
                       
                         ${\begin{array}{cl} \log (x) & if t = - 1 \\ \frac{x^{t + 1}}{t + 1} & if t \neq - 1 \end{array}$

By default,intreturns the general results for all values of the other symbolic parametert. In this example,intreturns two integral results for the case $t = - 1$ and $t \neq - 1$ .

To ignore special cases of parameter values, set'IgnoreSpecialCases'totrue. With this option,intignores the special case $t = - 1$ and returns the solution for $t \neq - 1$ .

F = int(x^t,x,'IgnoreSpecialCases',true)

F =
                       
                         $\frac{x^{t + 1}}{t + 1}$

Find Cauchy Principal Value

Open Live Script

Define a symbolic function $f (x) = 1 / (x - 1)$ that has a pole at $x = 1$ .

symsxf(x) = 1/(x-1)

f(x) =
                       
                         $\frac{1}{x - 1}$

Compute the definite integral of this function from $x = 0$ to $x = 2$ . Since the integration interval includes the pole, the result is not defined.

F = int (F (0 - 2))

F =
                        $NaN$

However, the Cauchy principal value of the integral exists. To compute the Cauchy principal value of the integral, set“PrincipalValue”totrue.

F = int(f,[0 2],“PrincipalValue”,true)

F =
                        $0$

Unevaluated Integral and Integration by Parts

Open Live Script

Find the integral of $\int x e^{x} dx$ .

Define the integral without evaluating it by setting the'Hold'option totrue.

symsxg(y)F = int(x*exp(x),'Hold',true)

F =
                       
                         $\int x e^{x} d x$

You can apply integration by parts toFby using theintegrateByPartsfunction. Useexp(x)as the differential to be integrated.

G = integrateByParts (F, exp (x))

G =
                       
                         $x e^{x} - \int e^{x} d x$

To evaluate the integral inG, use thereleasefunction to ignore the'Hold'option.

Gcalc = release(G)

Gcalc =
                        $x e^{x} - e^{x}$

Compare the result to the integration result returned byintwithout setting the'Hold'option.

Fcalc = int(x*exp(x))

Fcalc =
                        $e^{x} (x - 1)$

Approximate Indefinite Integrals

Open Live Script

Ifintcannot compute a closed form of an integral, then it returns an unresolved integral.

symsf(x)f(x) = sin(sinh(x)); F = int(f,x)

F(x) =
                       
                         $\int \sin (\sinh (x)) d x$

You can approximate the integrand function $f (x)$ as polynomials by using the Taylor expansion. Applytaylorto expand the integrand function $f (x)$ as polynomials around $x = 0$ . Compute the integral of the approximated polynomials.

fTaylor = taylor(f,x,'ExpansionPoint',0,'Order',10)

fTaylor(x) =
                       
                         $\frac{x^{9}}{5670} - \frac{x^{7}}{90} - \frac{x^{5}}{15} + x$

Fapprox = int(fTaylor,x)

Fapprox(x) =
                       
                         $\frac{x^{10}}{56700} - \frac{x^{8}}{720} - \frac{x^{6}}{90} + \frac{x^{2}}{2}$

Input Arguments

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`expr`—Integrand
symbolic expression|symbolic function|symbolic vector|symbolic matrix|symbolic number

Integrand, specified as a symbolic expression, function, vector, matrix, or number.

`var`—Integration variable
symbolic variable

Integration variable, specified as a symbolic variable. If you do not specify this variable,intuses the default variable determined bysymvar(expr,1). Ifexpris a constant, then the default variable isx.

`a`—Lower bound
number|symbolic number|symbolic variable|symbolic expression|symbolic function

Lower bound, specified as a number, symbolic number, variable, expression, or function (including expressions and functions with infinities).

`b`—Upper bound
number|symbolic number|symbolic variable|symbolic expression|symbolic function

Upper bound, specified as a number, symbolic number, variable, expression, or function (including expressions and functions with infinities).

Name-Value Arguments

Specify optional pairs of arguments asName1=Value1,...,NameN=ValueN, whereNameis the argument name andValueis the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and encloseNamein quotes.

Example:'IgnoreAnalyticConstraints',truespecifies thatintapplies purely algebraic simplifications to the integrand.

`IgnoreAnalyticConstraints`—Indicator for applying purely algebraic simplifications to integrand
`false`(default) |`true`

Indicator for applying purely algebraic simplifications to the integrand, specified astrueorfalse. If the value istrue, apply purely algebraic simplifications to the integrand. This option can provide simpler results for expressions, for which the direct use of the integrator returns complicated results. In some cases, it also enablesintto compute integrals that cannot be computed otherwise.

Using this option can lead to results not generally valid. This option applies mathematical identities that are convenient, but the results do not always hold for all values of variables. For details, seeAlgorithms.

`IgnoreSpecialCases`—Indicator for ignoring special cases
`false`(default) |`true`

Indicator for ignoring special cases, specified astrueorfalse. This ignores cases that require one or more parameters to be elements of a comparatively small set, such as a fixed finite set or a set of integers.

`PrincipalValue`—Indicator for returning principal value
`false`(default) |`true`

Indicator for returning the principal value, specified astrueorfalse. If the value istrue, compute the Cauchy principal value of the integral. In live script, the Cauchy principal value of unevaluated integral shows as thesymbol.

`Hold`—Indicator for unevaluated integration
`false`(default) |`true`

Indicator for unevaluated integration, specified astrueorfalse. If the value istrue,intreturns integrals without evaluating them.

提示s

In contrast to differentiation, symbolic integration is a more complicated task. Ifintcannot compute an integral of an expression, check for these reasons:
- The antiderivative does not exist in a closed form.
- The antiderivative exists, butint不能鳍d it.
Ifintcannot compute a closed form of an integral, it returns an unresolved integral.
Try approximating such integrals by using one of these methods:
- For indefinite integrals, use series expansions. Use this method to approximate an integral around a particular value of the variable.
- For definite integrals, use numeric approximations.
For indefinite integrals,intdoes not return a constant of integration in the result. The results of integrating mathematically equivalent expressions may be different. For example,syms x; int((x+1)^2)returns(x+1)^3/3, whilesyms x; int(x^2+2*x+1)returns(x*(x^2+3*x+3))/3, which differs from the first result by1/3.
For indefinite integrals,intimplicitly assumes that the integration variablevaris real. For definite integrals,intrestricts the integration variablevarto the specified integration interval. If one or both integration boundsaandbare not numeric,intassumes thata <= bunless you explicitly specify otherwise.

Algorithms

When you useIgnoreAnalyticConstraints,intapplies some of these rules:

log(a) + log(b) = log(a·b)for all values ofaandb. In particular, the following equality is valid for all values ofa,b,c:
(a·b)^c=a^c·b^c.
log(a^b) =b·log(a)for all values ofaandb. In particular, the following equality is valid for all values ofa,b,c:
(a^b)^c=a^b·c.
Iffandgare standard mathematical functions andf(g(x)) =xfor all small positive numbers, thenf(g(x)) =xis assumed to be valid for all complex valuesx. In particular:
- log(e^x) =x
- asin(sin(x)) =x,acos(cos(x)) =x,atan(tan(x)) =x
- asinh(sinh(x)) =x,acosh(cosh(x)) =x,atanh(tanh(x)) =x
- W_k(x·e^x) =xfor all branch indiceskof the LambertWfunction.

Version History

Introduced before R2006a

int

Syntax

Description

Examples

Indefinite Integral of Univariate Expression

Indefinite Integrals of Multivariate Function

Definite Integrals of Symbolic Expressions

Integrals of Matrix Elements

Apply IgnoreAnalyticConstraints

Ignore Special Cases

Find Cauchy Principal Value

Unevaluated Integral and Integration by Parts

Approximate Indefinite Integrals

Input Arguments

`expr`—Integrand
symbolic expression|symbolic function|symbolic vector|symbolic matrix|symbolic number

`var`—Integration variable
symbolic variable

`a`—Lower bound
number|symbolic number|symbolic variable|symbolic expression|symbolic function

`b`—Upper bound
number|symbolic number|symbolic variable|symbolic expression|symbolic function

Name-Value Arguments

`IgnoreAnalyticConstraints`—Indicator for applying purely algebraic simplifications to integrand
`false`(default) |`true`

`IgnoreSpecialCases`—Indicator for ignoring special cases
`false`(default) |`true`

`PrincipalValue`—Indicator for returning principal value
`false`(default) |`true`

`Hold`—Indicator for unevaluated integration
`false`(default) |`true`

提示s

Algorithms

Version History

See Also

Topics

int

Syntax

Description

Examples

Indefinite Integral of Univariate Expression

Indefinite Integrals of Multivariate Function

Definite Integrals of Symbolic Expressions

Integrals of Matrix Elements

Apply IgnoreAnalyticConstraints

Ignore Special Cases

Find Cauchy Principal Value

Unevaluated Integral and Integration by Parts

Approximate Indefinite Integrals

Input Arguments

expr—Integrandsymbolic expression|symbolic function|symbolic vector|symbolic matrix|symbolic number

var—Integration variablesymbolic variable

a—Lower boundnumber|symbolic number|symbolic variable|symbolic expression|symbolic function

b—Upper boundnumber|symbolic number|symbolic variable|symbolic expression|symbolic function

Name-Value Arguments

IgnoreAnalyticConstraints—Indicator for applying purely algebraic simplifications to integrandfalse(default) |true

IgnoreSpecialCases—Indicator for ignoring special casesfalse(default) |true

PrincipalValue—Indicator for returning principal valuefalse(default) |true

Hold—Indicator for unevaluated integrationfalse(default) |true

提示s

Algorithms

Version History

See Also

Topics

`expr`—Integrand
symbolic expression|symbolic function|symbolic vector|symbolic matrix|symbolic number

`var`—Integration variable
symbolic variable

`a`—Lower bound
number|symbolic number|symbolic variable|symbolic expression|symbolic function

`b`—Upper bound
number|symbolic number|symbolic variable|symbolic expression|symbolic function

`IgnoreAnalyticConstraints`—Indicator for applying purely algebraic simplifications to integrand
`false`(default) |`true`

`IgnoreSpecialCases`—Indicator for ignoring special cases
`false`(default) |`true`

`PrincipalValue`—Indicator for returning principal value
`false`(default) |`true`

`Hold`—Indicator for unevaluated integration
`false`(default) |`true`