Symbolic Integration Problem Using Symbolic Math Toolbox

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Dear Matlab users,
I would like to get the integral of a function symbolically in MATLAB. However, MATLAB doesn't give me the integral of this function. Do you have any advice?
Thanks in advance.
MATLAB Codes to run:
clear all; clc;
syms a q mL y nL p s b yA yU mR nR r
KK5=sin(q*pi*(mR*y+nR)/a)*cos((r*pi*(mR*y+nR))/a)*cos((2*p*pi*y)/b);
K5=simplify((-1/2)*((a*r)/(q^2-r^2)*pi)*int(KK5,y,[yA yU]))
It gives me this result:
K5 =
-(a*r*pi*int(cos((pi*r*(nR + mR*y))/a)*sin((pi*q*(nR + mR*y))/a)*cos((2*pi*p*y)/b), y, yA, yU))/(2*(q^2 - r^2))
  2 Comments
Walter Roberson
Walter Roberson on 31 Aug 2022
Integral with respect to which variable? You did not specify a variable of integration, so int() is going to pick one.
Note: when you syms i and syms j then the i and j that result will just be normal variables, with no connection to sqrt(-1) and no connection to coordinate axes. int() will not recognize hyperbolic functions, or possibilities of rewriting in terms of exp()
ercan duzgun
ercan duzgun on 31 Aug 2022
Thank you very much for your response. You are right to say that I didn't specify the variable of integration. I didn't notice at first. I have edited my codes according to your warnings. I also changed i and j variables to q and p variables.
However, I am still not receiving a normal integrated symbolic expression. Do you have any other advice?

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Accepted Answer

Torsten
Torsten on 31 Aug 2022
Edited: Torsten on 31 Aug 2022
syms par1 par2 par3 par4 par5
syms a q p s b mR nR r
syms y yA yU
K5 = cos(par1*y+par2)*sin(par3*y+par4)*cos(par5*y);
K55 = int(K5,y,yA,yU);
K55 = subs(K55,[par1 par2 par3 par4 par5],[q*pi*mR/a q*pi*nR/a r*pi*mR/a r*pi*nR/a 2*pi*p/b]);
K55 = (-1/2)*((a*r)/(q^2-r^2)*pi)*K55;
K55 = simplify(K55)
K55 = 
  9 Comments
Paul
Paul on 31 Aug 2022
It looks like a "divide by a" is missing in your expression for the argument of the sin(). For some reason, including that makes a big difference in the result. Still a mystery
syms a q mL y nL p s b yA yU mR nR r
% KK5=sin(q*pi*(mR*y+nR)) *cos(r*pi*(mR*y+nR)/a)*cos(2*p*pi*y/b);
KK5=sin(q*pi*(mR*y+nR)/a)*cos(r*pi*(mR*y+nR)/a)*cos(2*p*pi*y/b);
K5=(-1/2)*((a*r)/(q^2-r^2)*pi)*int(KK5,y,yA,yU);
simplify(K5)
ans = 
I wonder if there are some allowable values of the parameters (r,q, etc.) that make it not possible to evaluate the integral uniquely
Torsten
Torsten on 31 Aug 2022
It looks like a "divide by a" is missing in your expression for the argument of the sin().
Yes, that's why I wrote: Up to here, it works.

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More Answers (1)

ercan duzgun
ercan duzgun on 31 Aug 2022
Edited: ercan duzgun on 31 Aug 2022
Dear @Torsten, and dear @Paul, ( and dear @Walter Roberson)
I tried to follow @Torsten's last suggestion. Actually, I tried to use both method, however I get different results. Do you have any idea on "why I am getting different results " while using two different codes ?
My codes are like this:
clear all;clc;
%method1
syms a r iv mR nR y yA yU mL nL b jv iv
KK5=sin(iv*pi*(mR*y+nR)/a)*cos(r*pi*(mR*y+nR)/a)*cos(2*jv*pi*y/b)
KKK5=int(KK5,y,[yA yU])
KKKK5=simplify(((-1/2)*((a*r)/((iv^2-r^2)*pi)))*KKK5)
%method2
syms y5 p5 y55 p55 y555
Y5=sin(y5*y+p5)*cos(y55*y+p55)*cos(y555)
YY5=int(Y5,y,[yA yU])
YYY5=subs(YY5,[y5 p5 y55 p55 y555],[(iv*pi*mR/a) (iv*pi*nR/a) (r*pi*mR/a) (r*pi*nR/a) (2*jv*pi/b)])
YYYY5=simplify(((-1/2)*((a*r)/((iv^2-r^2)*pi)))*YYY5)
num_KKK5=eval(subs(KKKK5,[a r iv mR nR y yA yU mL nL b jv],[0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.25 0.35 0.45 0.55]))
num_YYYY5=eval(subs(YYYY5,[a r iv mR nR y yA yU mL nL b jv],[0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.25 0.35 0.45 0.55]))
simplify(KKKK5-YYYY5)
num_KKK5-num_YYYY5
%---
I get : num_KKK5 = -0.0016 , however, num_YYYY5 = -3.2468e-04 . Why different results?
Dear @Torsten , in your last message, you wrote to integrate directly. I can use your code to integrate directly, without using par1 etc. If I use your codes, it can directly integrate without any problem. However, when I tried to use my codes like that in the attachement below (method1), it can not integrate directly. It gives me result as :
KKKK5 =
-(a*r*int(cos((pi*r*(nR + mR*y))/a)*sin((pi*iv*(nR + mR*y))/a)*cos((2*pi*jv*y)/b), y, yA, yU))/(2*pi*(iv^2 - r^2))
  2 Comments
Torsten
Torsten on 31 Aug 2022
After correcting some errors in your code, I get the same result for both approaches.
format long
syms a r iv mR nR y yA yU mL nL b jv iv
KK5=sin(iv*pi*(mR*y+nR)/a)*cos(r*pi*(mR*y+nR)/a)*cos(2*jv*pi*y/b);
KKK5=int(KK5,y,yA,yU);
KKKK5=simplify(((-1/2)*((a*r)/((iv^2-r^2)*pi)))*KKK5);
num_KKK5=double(subs(KKKK5,[a r iv mR nR y yA yU mL nL b jv],[0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.25 0.35 0.45 0.55]))
num_KKK5 =
-0.001632012764608
%method2
syms y5 p5 y55 p55 y555
Y5=sin(y5*y+p5)*cos(y55*y+p55)*cos(y555*y);
YY5=int(Y5,y,yA,yU);
YYY5=simplify(((-1/2)*((a*r)/((iv^2-r^2)*pi)))*YY5);
YYYY5=subs(YYY5,[y5 p5 y55 p55 y555],[(iv*pi*mR/a) (iv*pi*nR/a) (r*pi*mR/a) (r*pi*nR/a) (2*jv*pi/b)]);
num_YYYY5=double(subs(YYYY5,[a r iv mR nR y yA yU mL nL b jv],[0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.25 0.35 0.45 0.55]))
num_YYYY5 =
-0.001632012764608
ercan duzgun
ercan duzgun on 31 Aug 2022
Dear @Torsten , thank you very much for your useful reply. I appreciate your response.
Kind regards,

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