MATLAB Tutor

Part 9: Integration

1. First we calculate indefinite integrals. Before doing anything, we must declare x to be a symbolic variable.   Enter:
   
   syms x
   
   Then enter
   
   int(x*sin(x), x)
   
   Check that the result is an antiderivative of x sin(x) by entering:
   
   diff(ans, x)
   
   


2. Introduce another symbolic variable a to investigate how MATLAB deals with more than one symbolic variable in integration. Enter:
   
   syms a
   
   Then enter
   
   int(a*sin(x), x)
   
   Now enter
   
   int(a*sin(x), a)
   
   What is the role of the variable following the comma?
   
   


3. Now try to find an antiderivative for sin ( x⁵ + x³ ).
   
   MATLAB does not know an antiderivative of this function that can be defined in terms of functions known to MATLAB. On the other hand, enter:
   
   int(sin(x^2), x)
   
   The Fresnel S function is known to MATLAB, but probably not to you. However, you can check by differentiation that the expression is an antiderivative.
   
   


4. Next we calculate definite integrals. To integrate x sin(x) over the interval [0,pi/2], enter:
   
   int(x*sin(x), x, 0, pi/2)
   
   


5. Now try this method on the integral of sin ( x⁵ + x³ ) over the interval [0,pi/2].
   
   MATLAB still doesn't know an antiderivative for this function. To obtain a numerical estimate, enter:
   
   double( int(sin(x^5 + x^3), x, 0, pi/2) )
   
   


6. If you know that all you want is a numerical estimate, you can use MATLAB's numerical integrator called quadl. (The last character is the letter "el".) No symbolics are involved. Enter:
   
   f=inline('sin(x^5+x^3)')
   quadl(f , 0, pi/2 )
   
   Numerical integration also works with user-defined functions in m-files. Symbolic integration doesn't. If you still have the m-file called fun.m that you created in Part 6 of this Tutorial, enter:
   
   quadl('fun', 0, pi/2 )
   
   The significance of using the quadl function is that MATLAB does not try to find a symbolic solution before starting on the numerical estimate.
   
   


7. Use MATLAB to find the exact value of each of the following integrals. (Type inf for the infinity symbol.)
   
   
   • The integral of 1/(1+x²) from 0 to 1,
     
     
   • The integral of 1/(1+x²) from 0 to Infinity,
     
     
   • The integral of 1/(1+x⁴) from 0 to Infinity.