definite and indefinite integrals -凯发k8网页登录

define a univariate expression.

syms x
expr = -2*x/(1 x^2)^2;

find the indefinite integral of the univariate expression.

f = int(expr)

f = 
1x2 1

define a multivariate function with variables x and z.

syms x z
f(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) = 
x22 z2 1

fz = int(f,z)

fz(x, z) = x atan(z)

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) = 
x22 z2 1

integrate a symbolic expression from 0 to 1.

syms x
expr = x*log(1 x);
f = int(expr,[0 1])

f = 
14

integrate another expression from sin(t) to 1.

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

f = cos(t)2

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

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

fint = 
∫010cos(x)x2 1 dx

fvpa = vpa(fint)

fvpa = 0.37570628299079723478493405557162

to approximate integrals directly, use instead of vpa. the vpaintegral function is faster and provides control over integration tolerances.

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

fvpaint = 0.375706

define a symbolic matrix containing four expressions as its elements.

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

m = 
(etea tsin(t)cos(t))

find indefinite integrals of the matrix element-wise.

f = int(m,t)

f = 
(etea ta-cos(t)sin(t))

define a symbolic function and compute its indefinite integral.

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

f(x) = 
x acos(cos(x))-x22 sign(sin(x))

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) = 
x22

define a symbolic expression xt and compute its indefinite integral with respect to the variable x.

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

f = 
{log(x) if t=-1xt 1t 1 if t≠-1

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≠-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≠-1.

f = int(x^t,x,'ignorespecialcases',true)

f = 
xt 1t 1

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

syms x
f(x) = 1/(x-1)

f(x) = 
1x-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' to true.

f = int(f,[0 2],'principalvalue',true)

f = 0

find the integral of ∫xexdx.

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 = 
∫x ex dx

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 ex-∫ex dx

to evaluate the integral in g, use the release function to ignore the 'hold' option.

gcalc = release(g)

gcalc = x ex-ex

compare the result to the integration result returned by int without setting the 'hold' option.

fcalc = int(x*exp(x))

fcalc = ex x-1

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) = 
∫sin(sinh(x)) dx

you can approximate the integrand function f(x) as polynomials by using the taylor expansion. apply 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) = 
x95670-x790-x515 x

fapprox = int(ftaylor,x)

fapprox(x) = 
x1056700-x8720-x690 x22