Symbolic Math Toolbox

Integration

This example shows how to compute definite and indefinite integrals using Symbolic Math Toolbox™.

Indefinite Integrals

To integrate a mathematical expression f means to find an expression F such that the first derivative of F is f. Integration is a more complicated task than differentiation. In contrast to differentiation, there is no general algorithm for computing integrals of an arbitrary expression. For example, create the following expression f:

syms x;
f = sin(x)/x
 
f =
 
sin(x)/x
 

You can visualize an expression by using the plot command. For example, create the plot of f for the values of variable x from -15 to 15:

ezplot(f, [-15, 15])

To compute integrals use the int function. When you integrate an expression, the result often involves much more complicated functions than those you use in the original expression. For example, the integral of a simple trigonometric expression sin(x)/x is a special function (the sine integral function):

F = int(f, x)
 
F =
 
sinint(x)
 

You also can visualize the sine integral function:

ezplot(F, [-15, 15])

Definite Integrals

The int function also allows you to compute definite integrals. To compute a definite integral, specify the upper and lower limits of the integration interval. For example, you can compute the integral of f = sin(x)/x for the interval from -15 to 15:

int(f, x, -15, 15)
 
ans =
 
2*sinint(15)
 

You also can use infinities when specifying one or both sides of the integration interval. For example, you can integrate the function f from 0 to positive infinity:

int(f, x, 0, inf)
 
ans =
 
pi/2
 

Integrating f from negative infinity to 0 gives the same result:

int(f, x, -inf, 0)
 
ans =
 
pi/2
 

You can also compute the integral of f over the set of all real numbers:

int(f, x, -inf, inf)
 
ans =
 
pi