If f
is a symbolic expression, then
int(f)
attempts to find another symbolic expression, F
,
so that diff(F)
=
f
. That is, int(f)
returns
the indefinite integral or antiderivative of f
(provided
one exists in closed form). Similar to differentiation,
int(f,v)
uses the symbolic object v
as the variable
of integration, rather than the variable determined by symvar
.
See how int
works by looking at this table.
Mathematical Operation  MATLAB^{®} Command 

$$\int {x}^{n}}dx=\{\begin{array}{ll}\mathrm{log}(x)\hfill & \text{if}n=1\hfill \\ \frac{{x}^{n+1}}{n+1}\hfill & \text{otherwise}\text{.}\hfill \end{array$$ 

$$\underset{0}{\overset{\pi /2}{\int}}\mathrm{sin}(2x)dx=1$$ 

g = cos(at + b) $$\int g(t)dt=\mathrm{sin}(at+b)/a$$ 

$$\int {J}_{1}(z)dz={J}_{0}(z)$$ 

In contrast to differentiation, symbolic integration is a more complicated task. A number of difficulties can arise in computing the integral:
The antiderivative, F
, may not
exist in closed form.
The antiderivative may define an unfamiliar function.
The antiderivative may exist, but the software can't find it.
The software could find the antiderivative on a larger computer, but runs out of time or memory on the available machine.
Nevertheless, in many cases, MATLAB can perform symbolic integration successfully. For example, create the symbolic variables
syms a b theta x y n u z
The following table illustrates integration of expressions containing those variables.
f  int(f) 

syms x n f = x^n;  int(f) ans = piecewise(n == 1, log(x), n ~= 1,... x^(n + 1)/(n + 1)) 
syms y f = y^(1);  int(f) ans = log(y) 
syms x n f = n^x;  int(f) ans = n^x/log(n) 
syms a b theta f = sin(a*theta+b);  int(f) ans = cos(b + a*theta)/a 
syms u f = 1/(1+u^2);  int(f) ans = atan(u) 
syms x f = exp(x^2);  int(f) ans = (pi^(1/2)*erf(x))/2 
In the last example, exp(x^2)
, there is
no formula for the integral involving standard calculus expressions,
such as trigonometric and exponential functions. In this case, MATLAB returns
an answer in terms of the error function erf
.
If MATLAB is unable to find an answer to the integral of
a function f
, it just returns int(f)
.
Definite integration is also possible.
Definite Integral  Command 

$${\int}_{a}^{b}f(x)dx$$ 

$${\int}_{a}^{b}f(v)dv$$ 

Here are some additional examples.
f  a, b  int(f, a, b) 

syms x f = x^7;  a = 0; b = 1;  int(f, a, b) ans = 1/8 
syms x f = 1/x;  a = 1; b = 2;  int(f, a, b) ans = log(2) 
syms x f = log(x)*sqrt(x);  a = 0; b = 1;  int(f, a, b) ans = 4/9 
syms x f = exp(x^2);  a = 0; b = inf;  int(f, a, b) ans = pi^(1/2)/2 
syms z f = besselj(1,z)^2;  a = 0; b = 1;  int(f, a, b) ans = hypergeom([3/2, 3/2],... [2, 5/2, 3], 1)/12 
For the Bessel function (besselj
)
example, it is possible to compute a numerical approximation to the
value of the integral, using the double
function.
The commands
syms z a = int(besselj(1,z)^2,0,1)
return
a = hypergeom([3/2, 3/2], [2, 5/2, 3], 1)/12
and the command
a = double(a)
returns
a = 0.0717
One of the subtleties involved in symbolic integration is the “value” of various parameters. For example, if a is any positive real number, the expression
$${e}^{a{x}^{2}}$$
is the positive, bell shaped curve that tends to 0 as x tends
to ±∞. You can
create an example of this curve, for a = 1/2. Prior to
R2016a, use ezplot
instead of fplot
.
syms x
a = sym(1/2);
f = exp(a*x^2);
fplot(f)
However, if you try to calculate the integral
$$\underset{\infty}{\overset{\infty}{\int}}{e}^{a{x}^{2}}}dx$$
without assigning a value to a, MATLAB assumes
that a represents a complex number, and therefore
returns a piecewise answer that depends on the argument of a.
If you are only interested in the case when a is
a positive real number, use assume
to set an assumption
on a
:
syms a assume(a > 0)
Now you can calculate the preceding integral using the commands
syms x f = exp(a*x^2); int(f, x, inf, inf)
This returns
ans = pi^(1/2)/a^(1/2)
To calculate the integral
$$\underset{\infty}{\overset{\infty}{\int}}\frac{1}{{a}^{2}+{x}^{2}}\text{\hspace{0.05em}}dx$$
for complex values of a
, enter
syms a x f = 1/(a^2 + x^2); F = int(f, x, inf, inf)
Use syms
to clear the all assumptions on variables. For more information
about symbolic variables and assumptions on them, see Delete Symbolic Objects and Their Assumptions.
The preceding commands produce the complex output
F = (pi*signIm(1i/a))/a
The function signIm
is defined as:
$$\text{signIm}(z)=\{\begin{array}{ll}1\hfill & \text{if}\mathrm{Im}(z)0,\text{or}\mathrm{Im}(z)=0\text{and}z0\hfill \\ 0\hfill & \text{if}z=0\hfill \\ \text{1}\hfill & \text{otherwise}\text{.}\hfill \end{array}$$
To evaluate F
at a = 1 + i
,
enter
g = subs(F, 1 + i)
g = pi*(1/2  1i/2)
double(g)
ans = 1.5708  1.5708i
Highprecision numerical integration is implemented in the vpaintegral
function of the Symbolic Math
Toolbox™. vpaintegral
uses
variableprecision arithmetic in contrast to the MATLAB integral
function, which uses doubleprecision
arithmetic.
Integrate besseli(5,25*u).*exp(u*25)
by
using both integral
and vpaintegral
.
The integral
function returns NaN
and
issues a warning while vpaintegral
returns the
correct result.
syms u f = besseli(5,25*x).*exp(x*25); fun = @(u)besseli(5,25*u).*exp(u*25); usingIntegral = integral(fun, 0, 30) usingVpaintegral = vpaintegral(f, 0, 30)
Warning: Infinite or NotaNumber value encountered. usingIntegral = NaN usingVpaintegral = 0.688424
For more information, see vpaintegral
.