Symbolic Integration Problem Using Symbolic Math Toolbox

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))

[Walter Roberson]

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]

Dear Mr. @Walter Roberson;

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?

Answer 1

[Torsten]

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 = 

[ercan duzgun]

Dear @Torsten, thank you very much. I didn't think like that before. Your response is very useful. I appreciate your response and help.

[Torsten]

Of course, special cases are excluded, e.g. a=0, b=0, q^2=r^2.

[Paul]

Are all of those substitutions correct? It looks like par1 and par2 should be flipped with par3 and par4.

Any idea why using par1-5 and then subsing yields a closed form expression? That seems like a handy trick to remember.

[ercan duzgun]

Dear @Paul, you are right. Those substitutions are not correct. I noticed that they are not correct. However, the methodology to get the integral is useful. I can edit the wrong subsitutions into correct form by myself. Thank you for your warning.

[Torsten]

Better flip "sin" and "cos" in K5 :-)

[Paul]

@Torsten

Any insight into why using the par variables allowed int to find a closed from solution, when it couldn't using the original variables? That was very interesting to me.

[Torsten]

Up to here, it works.

One of the new big mysteries of the symbolic toolbox.

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);
K5=(-1/2)*((a*r)/(q^2-r^2)*pi)*int(KK5,y,yA,yU);
simplify(K5)

ans = 

[Paul]

@Torsten,

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]

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.

Answer 2

[ercan duzgun]

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))

[Torsten]

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]

Dear @Torsten , thank you very much for your useful reply. I appreciate your response.

Kind regards,