Symbolic Math Toolbox

Integration

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 findsym. See how int works by looking at this table.

Mathematical Operation	MATLAB Command
	int(x^n) or int(x^n,x)
	int(sin(2*x),0,pi/2) or int(sin(2*x),x,0,pi/2)
	g = cos(a*t + b) int(g) or int(g,t)
	int(besselj(1,z)) or int(besselj(1,z),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 the 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)
x^n	x^(n+1)/(n+1)
y^(-1)	log(y)
n^x	1/log(n)*n^x
sin(a*theta+b)	-1/a*cos(a*theta+b)
1/(1+u^2)	atan(u)
exp(-x^2)	1/2*pi^(1/2)*erf(x)

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. The commands

• int(f,a,b)
  

and

• int(f,v,a,b)
  

are used to find a symbolic expression for

respectively.

Here are some additional examples.

f	a, b	int(f,a,b)
x^7	0, 1	1/8
1/x	1, 2	log(2)
log(x)*sqrt(x)	0, 1	-4/9
exp(-x^2)	0, inf	1/2*pi^(1/2)
besselj(1,z)^2	0, 1	1/12*hypergeom([3/2, 3/2], [2, 5/2, 3],-1)

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 =
  1/12*hypergeom([3/2, 3/2],[2, 5/2, 3],-1)
  

and the command

• a = double(a)
  

returns

• a =
    0.0717
  

Integration with Real Parameters

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

is the positive, bell shaped curve that tends to 0 as x tends to . You can create an example of this curve, for , using the following commands:

• syms x
  a = sym(1/2);
  f = exp(-a*x^2);
  ezplot(f)
  
  
  

However, if you try to calculate the integral

without assigning a value to a, MATLAB assumes that a represents a complex number, and therefore returns a complex answer. If you are only interested in the case when a is a positive real number, you can calculate the integral as follows:

• syms a positive;
  

The argument positive in the syms command restricts a to have positive values. 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 =
  1/(a)^(1/2)*pi^(1/2)
  

If you want to calculate the integral

for any real number a, not necessarily positive, you can declare a to be real with the following commands:

• syms a real
  f=exp(-a*x^2);
  F = int(f, x, -inf, inf)
  

MATLAB returns

• F =
  PIECEWISE([1/a^(1/2)*pi^(1/2), signum(a) = 1],[Inf, otherwise])
  

You can put this in a more readable form by entering

• pretty(F)
                           {   1/2
                           { pi
                           { -----        signum(a~) = 1
                           {   1/2
                           { a~
                           {
                           {  Inf           otherwise
  

The ~ after a is simply a reminder that a is real, and signum(a~) is the sign of a. So the integral is

when a is positive, just as in the preceding example, and when a is negative.

You can also declare a sequence of symbolic variables w, y, x, z to be real by entering

• syms w x y z real
  

Integration with Complex Parameters

To calculate the integral

for complex values of a, enter

• syms a x unreal % 
  f = exp(-a*x^2);
  F = int(f, x, -inf, inf)
  

Note that syms is used with the unreal option to clear the real property that was assigned to a in the preceding example -- see Clearing Variables in the Maple Workspace.

The preceding commands produce is the complex output

• F =
  PIECEWISE([1/a^(1/2)*pi^(1/2), csgn(a) = 1],[Inf, otherwise])
  

You can make this output more readable by entering

• pretty(F)
                            {   1/2
                            { pi
                            { -----        csgn(a) = 1
                            {  1/2
                            { a
                            {
                            {  Inf          otherwise
  

The expression csgn(a) (complex sign of a) is defined by

The condition csgn(a) = 1 corresponds to the shaded region of the complex plane shown in the following figure.

The square root of a in the answer is the unique square root lying in the shaded region.

Limits

Symbolic Summation