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The difference between 'int' and 'integral' in terms of error (in case definite integral)?

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work wolf
work wolf on 12 Jun 2017
Commented: Nithin M on 21 Jul 2018
can i use 'integral' for an analytic integration in case definite integral compare with using 'int'? because when using 'int ' taken time to implement a program.
please,help me, i would like to know about
  1. error between them ,
  2. is there an equivalent code for 'int' and
  3. is 'quad' or ' integral' sufficient. What do you advise me to avoid the accumulated error where it is used several times in the same code?
  4. Can i Implement ' int' to evaluate int(cos(pi*x).^n,x) , n>=1, by matlab.
This is my code.
>> syms x n
>> int(cos(pi*x).^n,x, -10,10)
  • This is what I'm getting by MATLAB 2013 a
Warning: Explicit integral could not be found.
ans =
int(cos(pi*x)^n, x == -10..10)

Accepted Answer

John BG
John BG on 18 Jun 2017
Hi Work Wolf
1.
command int is for symbolic, to just obtain the primitive, which is what you are after,
int(expr,var)
or definite symbolic integral
int(expr,var,a,b)
command integral is for numerical integration
2.
having said this, int doesn't work for certain expressions, it's a limitation directly mention in MATLAB help. One tries
syms x;int(sin(sinh(x)),x)
= int(sin(sinh(x)), x)
the displayed answer is the same trivial input, like your function (cos(x))^n
3.
but there's a way around
F=(cos(x))^n;int(taylor(F,x,'ExpansionPoint',0,'Order',10),x)
F =
(n^4/3456 - n^3/864 + (7*n^2)/4320 - (17*n)/22680)*x^9 + (- n^3/336 + n^2/168 - n/315)*x^7 + (n^2/40 - n/60)*x^5 - (n*x^3)/6 + x
an approximation of the primitive for (cos(x))^n has been obtained.
4. Now the approximation error:
Please note that you are already in control of the error because you decide how long the input taylor development is, in the field 'Order'.
to measure the error after the integral, turn x and n into numerical values, integrate and obtain the difference with the exact expression
I don't know if you only want n integer, or you may want to check n real, (or complex and then measure for instance real and imaginary integrals)
perhaps the following is enough for the error checks, the following lines can be implemented with a for loop, but don't know the range of n and x you want.
when n=1
n=1;
x=[0:1/10:2*pi];F1=(cos(x)).^n;A0=integral(@(x)(cos(x)).^n,0,pi/2)
A0 = 1.0000
x1=0;A1= (n^4/3456 - n^3/864 + (7*n^2)/4320 - (17*n)/22680)*x1^9 + (- n^3/336 + n^2/168 - n/315)*x1^7 + (n^2/40 - n/60)*x1^5 - (n*x1^3)/6 + x1
A1 = 0
x2=pi/2;A2= (n^4/3456 - n^3/864 + (7*n^2)/4320 - (17*n)/22680)*x2^9 + (- n^3/336 + n^2/168 - n/315)*x2^7 + (n^2/40 - n/60)*x2^5 - (n*x2^3)/6 + x2
A2 = 1.0000
err1=A0-(A2-A1)
err1 = -3.5426e-06
when n=2
n=2;
F2=(cos(x)).^n;A0=integral(@(x)(cos(x)).^n,0,pi/2)
x1=0;A1= (n^4/3456 - n^3/864 + (7*n^2)/4320 - (17*n)/22680)*x1^9 + (- n^3/336 + n^2/168 - n/315)*x1^7 + (n^2/40 - n/60)*x1^5 - (n*x1^3)/6 + x1
x2=pi/2;A2= (n^4/3456 - n^3/864 + (7*n^2)/4320 - (17*n)/22680)*x2^9 + (- n^3/336 + n^2/168 - n/315)*x2^7 + (n^2/40 - n/60)*x2^5 - (n*x2^3)/6 + x2
err1=A0-(A2-A1)
A0 = 0.7854
A1 = 0
A2 = 0.7871
err1 = -0.0017
when n=3
n=3;
F2=(cos(x)).^n;A0=integral(@(x)(cos(x)).^n,0,pi/2)
x1=0;A1= (n^4/3456 - n^3/864 + (7*n^2)/4320 - (17*n)/22680)*x1^9 + (- n^3/336 + n^2/168 - n/315)*x1^7 + (n^2/40 - n/60)*x1^5 - (n*x1^3)/6 + x1
x2=pi/2;A2= (n^4/3456 - n^3/864 + (7*n^2)/4320 - (17*n)/22680)*x2^9 + (- n^3/336 + n^2/168 - n/315)*x2^7 + (n^2/40 - n/60)*x2^5 - (n*x2^3)/6 + x2
err1=A0-(A2-A1)
A0 = 0.6667
A1 = 0
A2 = 0.7130
err1 = -0.0463
5. It's also worth mentioning that if you only want the primitive with symbolic x, that it's ok to sweep n, then a verbose but nevertherless valid primitive is obtained expanding the product:
syms x n
int(cos(x)*cos(x),x)
= x/2+sin(2*x)/4
int(cos(x)*cos(x)*cos(x),x)
=sin(x)-sin(x)^3/3
int(cos(x)*cos(x)*cos(x)*cos(x),x)
=(3*x)/8+sin(2*x)/4+sin(4*x)/32
this primitives, without symbolic n, are exact.
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thanks in advance
John BG
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