How to calculate integrals in Matlab / Octave
To calculate the integral of a function f (x) in Matlab and Octave use int() function
int(f,x,inf,sup)
The parameters of the function are
- f is the function
- x is the integration variable
- inf is the lower extreme of integration
- sup is the upper extreme of integration
If the extremes of integration (inf, sup) are given, the function int () calculates a definite integral in the integration interval.
$$ \int_{inf}^{sup} f(x) \ dx $$
If the integration extremes are not indicated, the int() function calculates a indefinite integral of the function f(x).
$$ \int f(x) \ dx $$
Note. In Octave the int() function requires the installation and loading of the Symbolic library.
Examples
Example 1 (indefinite integral)
Define the variable x of the function as a symbol using symbolic
syms x
Define the function f (x) = x2
f=x**2
Integrate the function with the int() function using the x integration variable
int(f)
The output result is the indefinite integral of the function
ans = (sym) x^3/3
The indefinite integral of f(x)=x2 with respect to the variable x is x3/3
$$ \int x^2 \ dx = \frac{x^3}{3} + c $$
Example 2 (function of two variables)
Define two variables x and y
syms x y
Define a function with two variables f(x,y)=x*y
f=x*y
Integrate the function with respect to the variable y
int(f,y)
The output result is the integral
ans = (sym) x*y^2/2
The indefinite integral of f(x, y)=xy with respect to the variable y is xy2/2
$$ \int x \cdot y \ dx = x \cdot \frac{y^2}{2} + c $$
Example 3 (definite integral)
Define a variable x
sym x
Define the function f(x)=x+1
f = x+1
Compute definite integral of the function with respect to the variable x using the extremes of integration inf = 1 and sup = 3
int(f,x,1,3)
The result is the definite integral of the function f (x) in the interval (1,3)
ans = (sym) 6
For a quick check
$$ \int_1^3 x+1 \ dx = [ \frac{x^2}{2} + x ]^3_1 = $$
$$ = \frac{3^2}{2} +3 - \frac{1^2}{2} - 1 $$
$$ = \frac{9+6-1-2}{2} $$
$$ = \frac{12}{2}=6 $$
