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int

Definite and indefinite integrals

Syntax

int(expr,var)
int(expr,var,a,b)
int(___,Name,Value)

Description

example

int(expr,var) computes the indefinite integral of expr with respect to the symbolic scalar variable var. Specifying the variable var is optional. If you do not specify it, int uses the default variable determined by symvar. If expr is a constant, then the default variable is x.

example

int(expr,var,a,b) computes the definite integral of expr with respect to var from a to b. If you do not specify it, int uses the default variable determined by symvar. If expr is a constant, then the default variable is x.

int(expr,var,[a b]) is equivalent to int(expr,var,a,b).

example

int(___,Name,Value) specifies additional options using one or more Name,Value pair arguments. For example, 'IgnoreAnalyticConstraints',true specifies that int applies additional simplifications to the integrand.

Examples

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Find an indefinite integral of this univariate expression.

syms x
f = -2*x/(1+x^2)^2;
int(f)
ans =
1/(x^2 + 1)

Find indefinite integrals of this multivariate expression with respect to the variables x and z.

syms x z
f = x/(1+z^2);
int(f,x)
int(f,z)
ans =
x^2/(2*(z^2 + 1))
 
ans =
x*atan(z)

If you do not specify the integration variable, int uses the variable returned by symvar. For this expression, symvar returns x.

symvar(f, 1)
ans =
x

Integrate an expression from 0 to 1.

syms x
f = x*log(1+x);
int(f,[0 1])
ans =
1/4

Integrate an expression from sin(t) to 1.

syms t
int(2*x, [sin(t) 1])
ans =
cos(t)^2

When int cannot compute a definite integral, numerically approximate the integral by using vpa.

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

fVpa = vpaintegral(f,x,[0 10])
fVpa =
0.375706

Find indefinite integrals for the expressions listed as the elements of a matrix.

syms a x t z
M = [exp(t) exp(a*t); sin(t) cos(t)];
int(M)
ans =
[  exp(t), exp(a*t)/a]
[ -cos(t),     sin(t)]

Compute this indefinite integral. By default, int uses strict mathematical rules. These rules do not let int rewrite asin(sin(x)) and acos(cos(x)) as x.

syms x
f = acos(sin(x));
int(f,x)
ans =
x*acos(sin(x)) + x^2/(2*sign(cos(x)))

If you want a simple practical solution, try IgnoreAnalyticConstraints.

int(f, x, 'IgnoreAnalyticConstraints', true)
ans =
-(x*(x - pi))/2

Compute this integral with respect to the variable x. By default, int returns the integral as a piecewise object where every branch corresponds to a particular value (or a range of values) of the symbolic parameter t.

syms x t
int(x^t, x)
ans =
piecewise(t == -1, log(x), t ~= -1, x^(t + 1)/(t + 1))

To ignore special cases of parameter values, use IgnoreSpecialCases. With this option, int ignores the special case t=-1 and returns only the branch where t<>–1.

int(x^t, x, 'IgnoreSpecialCases', true)
ans =
x^(t + 1)/(t + 1)

Compute this definite integral, where the integrand has a pole in the interior of the interval of integration. Mathematically, this integral is not defined.

syms x
f = 1/(x-1);
int(f,x,0,2)
ans =
NaN

However, the Cauchy principal value of the integral exists. Use PrincipalValue to compute the Cauchy principal value of the integral.

int(f,x,0,2,'PrincipalValue',true)
ans =
0

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

syms x
f = sin(sinh(x));
int(f,x)
ans =
int(sin(sinh(x)), x)

If int cannot compute a closed form of an indefinite integral, try to approximate the expression around some point using taylor, and then compute the integral. For example, approximate the expression around x = 0.

fApprox = taylor(f, x, 'ExpansionPoint', 0, 'Order', 10);
int(fApprox,x)
ans =
x^10/56700 - x^8/720 - x^6/90 + x^2/2

Input Arguments

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Integrand, specified as a symbolic expression or function, a constant, or a vector or matrix of symbolic expressions, functions, or constants.

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.

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

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

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: 'IgnoreAnalyticConstraints',true specifies that int applies purely algebraic simplifications to the integrand.

Indicator for applying purely algebraic simplifications to 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. Sometimes, it also enables int to compute integrals that cannot be computed otherwise.

Using this option can lead to wrong or incomplete results.

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.

Indicator for returning principal value, specified as true or false. If the value is true, compute the Cauchy principal value of the integral.

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.

    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.

  • Results returned by int do not include integration constants.

  • 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 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 = ac·bc.

  • log(ab) = 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:

      (ab)c = ab·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(ex) = 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

    • lambertWk(x·ex) = x for all values of k

Introduced before R2006a