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

computes the indefinite integral of `F`

= int(expr)`expr`

. `int`

uses the default integration variable determined by symvar(`expr,1`

). If `expr`

is a constant, then the default integration variable is `x`

.

example

computes the indefinite integral of `F`

= int(expr,var)`expr`

with respect to the symbolic scalar variable `var`

.

example

computes the definite integral of `F`

= int(expr,a,b)`expr`

from `a`

to `b`

. `int`

uses the default integration variable determined by symvar(`expr,1`

). If `expr`

is a constant, then the default integration variable is `x`

.

`int(expr,[a b])`

is equivalent to `int(expr,a,b)`

.

example

computes the definite integral of `F`

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

with respect to the symbolic scalar variable `var`

from `a`

to `b`

.

`int(expr,var,[a b])`

is equivalent to `int(expr,var,a,b)`

.

example

specifies additional options using one or more `F`

= int(___,Name,Value)`Name,Value`

pair arguments. For example, `'IgnoreAnalyticConstraints',true`

specifies that `int`

applies additional simplifications to the integrand.

**Note**

The `int`

function computes integral symbolically, and it is not related to integer data types in MATLAB^{®}. For more information about integers, see Integers.

## Examples

collapse all

### Indefinite Integral of Univariate Expression

Open Live Script

Define a univariate expression.

`syms xexpr = -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 variables `x`

and `z`

.

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

Find the indefinite integrals of the multivariate expression with respect to the variables `x`

and `z`

.

Fx = int(f,x)

Fx(x, z) =$$\frac{{x}^{2}}{2\hspace{0.17em}\left({z}^{2}+1\right)}$$

Fz = int(f,z)

`Fz(x, z) =$$x\hspace{0.17em}\mathrm{atan}\left(z\right)$$`

If you do not specify the integration variable, then `int`

uses the first variable returned by `symvar`

as the integration variable.

var = symvar(f,1)

`var =$$x$$`

F = int(f)

F(x, z) =$$\frac{{x}^{2}}{2\hspace{0.17em}\left({z}^{2}+1\right)}$$

### Definite Integrals of Symbolic Expressions

Open Live Script

Integrate a symbolic expression from `0`

to `1`

.

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

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

Integrate another expression from `sin(t)`

to `1`

.

`syms tF = int(2*x,[sin(t) 1])`

`F =$${\mathrm{cos}\left(t\right)}^{2}$$`

When `int`

cannot compute the value of a definite integral, numerically approximate the integral by using vpa.

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

Fint =$${\int}_{0}^{10}\frac{\mathrm{cos}\left(x\right)}{\sqrt{{x}^{2}+1}}\mathrm{}\mathrm{d}x$$

Fvpa = vpa(Fint)

`Fvpa =$$0.37570628299079723478493405557162$$`

To approximate integrals directly, use vpaintegral instead of `vpa`

. The `vpaintegral`

function 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.

syms a x t zM = [exp(t) exp(a*t); sin(t) cos(t)]

M =$$\left(\begin{array}{cc}{\mathrm{e}}^{t}& {\mathrm{e}}^{a\hspace{0.17em}t}\\ \mathrm{sin}\left(t\right)& \mathrm{cos}\left(t\right)\end{array}\right)$$

Find indefinite integrals of the matrix element-wise.

F = int(M,t)

F =$$\left(\begin{array}{cc}{\mathrm{e}}^{t}& \frac{{\mathrm{e}}^{a\hspace{0.17em}t}}{a}\\ -\mathrm{cos}\left(t\right)& \mathrm{sin}\left(t\right)\end{array}\right)$$

### Apply IgnoreAnalyticConstraints

Open Live Script

Define a symbolic function and compute its indefinite integral.

`syms f(x)f(x) = acos(cos(x));F = int(f,x)`

F(x) =$$x\hspace{0.17em}\mathrm{acos}\left(\mathrm{cos}\left(x\right)\right)-\frac{{x}^{2}}{2\hspace{0.17em}\mathrm{sign}\left(\mathrm{sin}\left(x\right)\right)}$$

By default, `int`

uses strict mathematical rules. These rules do not let `int`

rewrite `acos(cos(x))`

as `x`

.

If you want a simple practical solution, set `'IgnoreAnalyticConstraints'`

to `true`

.

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

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

### Ignore Special Cases

Open Live Script

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

syms x tF = int(x^t,x)

F =$$\{\begin{array}{cl}\mathrm{log}\left(x\right)& \text{if}t=-1\\ \frac{{x}^{t+1}}{t+1}& \text{if}t\ne -1\end{array}$$

By default, `int`

returns the general results for all values of the other symbolic parameter `t`

. In this example, `int`

returns two integral results for the case $$t=-1$$ and $$t\ne -1$$.

To ignore special cases of parameter values, set `'IgnoreSpecialCases'`

to `true`

. With this option, `int`

ignores the special case $$t=-1$$ and returns the solution for $$t\ne -1$$.

`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 $$x=1$$.

`syms xf(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 =$$\mathrm{NaN}$$`

However, the Cauchy principal value of the integral exists. To compute the Cauchy principal value of the integral, set `'PrincipalValue'`

to `true`

.

`F = int(f,[0 2],'PrincipalValue',true)`

`F =$$0$$`

### Unevaluated Integral and Integration by Parts

Open Live Script

Find the integral of $\int \mathit{x}\text{\hspace{0.17em}}{\mathit{e}}^{\mathit{x}}\text{\hspace{0.17em}}\mathit{dx}$.

Define the integral without evaluating it by setting the `'Hold'`

option to `true`

.

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

F =$$\int x\hspace{0.17em}{\mathrm{e}}^{x}\mathrm{}\mathrm{d}x$$

You can apply integration by parts to `F`

by using the `integrateByParts`

function. Use `exp(x)`

as the differential to be integrated.

G = integrateByParts(F,exp(x))

G =$$x\hspace{0.17em}{\mathrm{e}}^{x}-\int {\mathrm{e}}^{x}\mathrm{}\mathrm{d}x$$

To evaluate the integral in `G`

, use the `release`

function to ignore the `'Hold'`

option.

Gcalc = release(G)

`Gcalc =$$x\hspace{0.17em}{\mathrm{e}}^{x}-{\mathrm{e}}^{x}$$`

Compare the result to the integration result returned by `int`

without setting the `'Hold'`

option.

Fcalc = int(x*exp(x))

`Fcalc =$${\mathrm{e}}^{x}\hspace{0.17em}\left(x-1\right)$$`

### Approximate Indefinite Integrals

Open Live Script

If `int`

cannot compute a closed form of an integral, then it returns an unresolved integral.

`syms f(x)f(x) = sin(sinh(x));F = int(f,x)`

F(x) =$$\int \mathrm{sin}\left(\mathrm{sinh}\left(x\right)\right)\mathrm{}\mathrm{d}x$$

You can approximate the integrand function $$f(x)$$ as polynomials by using the Taylor expansion. Apply taylor to 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, `int`

uses the default variable determined by `symvar(expr,1)`

. If expr is a constant, then the default variable is `x`

.

`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 as `Name1=Value1,...,NameN=ValueN`

, where `Name`

is the argument name and `Value`

is 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 enclose* `Name`

*in quotes.*

**Example: **`'IgnoreAnalyticConstraints',true`

specifies that `int`

applies 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 as `true`

or `false`

. If the value is `true`

, 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 enables `int`

to 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, see Algorithms.

`IgnoreSpecialCases`

— Indicator for ignoring special cases

`false`

(default) | `true`

Indicator for ignoring special cases, specified as `true`

or `false`

. 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 as `true`

or `false`

. If the value is `true`

, `int`

computes the Cauchy principal value of the integral. In live scripts, the Cauchy principal value of the unevaluated integral appears as the symbol.

`Hold`

— Indicator for unevaluated integration

`false`

(default) | `true`

Indicator for unevaluated integration, specified as `true`

or `false`

. If the value is `true`

, `int`

returns integrals without evaluating them.

## Tips

In contrast to differentiation, symbolic integration is a more complicated task. If

`int`

cannot compute an integral of an expression, check for these reasons:The antiderivative does not exist in a closed form.

The antiderivative exists, but

`int`

cannot find it.

If

`int`

cannot compute a closed form of an integral, it returns an unresolved integral.For some integrals that have closed form solutions, where these solutions are complicated and

`int`

returns unresolved integrals, you can use`simplify`

to obtain the closed form solutions. For example, the following code finds the closed form solution of the integral of`f(x)`

:`syms xf(x) = x*log(x/2+sqrt(x^2+1));F = int(f,x)simplify(F,Steps=10)`

Otherwise, you can try approximating unresolved 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,

`int`

does 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`

, while`syms x; int(x^2+2*x+1)`

returns`(x*(x^2+3*x+3))/3`

, which differs from the first result by`1/3`

.For indefinite integrals,

`int`

implicitly assumes that the integration variable var is real. For definite integrals,`int`

restricts the integration variable`var`

to the specified integration interval. If one or both integration bounds`a`

and`b`

are not numeric,`int`

assumes that`a <= b`

unless you explicitly specify otherwise.

## Algorithms

When you use `IgnoreAnalyticConstraints`

, `int`

applies some of these rules:

log(

*a*) + log(*b*)=log(*a*·*b*) for all values of*a*and*b*. In particular, the following equality is valid for all values of*a*,*b*, and*c*:(

*a*·*b*)^{c}=*a*^{c}·*b*^{c}.log(

*a*^{b})=*b*·log(*a*) for all values of*a*and*b*. In particular, the following equality is valid for all values of*a*,*b*, and*c*:(

*a*^{b})^{c}=*a*^{b·c}.If

*f*and*g*are standard mathematical functions and*f*(*g*(*x*))=*x*for all small positive numbers, then*f*(*g*(*x*))=*x*is assumed to be valid for all complex values*x*. 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})=*x*for all branch indices*k*of the Lambert W function.

## Version History

**Introduced before R2006a**

expand all

### R2019b: Return unevaluated integral

`int(___,'Hold',true)`

returns integrals without evaluating them. Use release to return the evaluated integrals by ignoring the `'Hold'`

option in the `int`

function.

## See Also

diff | dsolve | functionalDerivative | symvar | vpaintegral | integrateByParts | changeIntegrationVariable | release | rewrite

### Topics

- Integration

### External Websites

- Calculus Integrals (MathWorks Teaching Resources)
- Beam Bending and Deflection (MathWorks Teaching Resources)

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