Integration of a function which has limits in terms of parameters

Hi,

I have a function f3(t), defined as

clc; clear all;
syms  t L rho m L n T  k  G  v

N(t) = (T*L/2)*(n*pi/L)^2+k*sin(n*pi*v*t/L);
D(t) = (rho*L/2+m*(sin(n*pi*v*t/L))^2);
alpha(t) = N/D;
i = 6; j = 1
f3(t) = alpha(t)*cos(2*pi*i*t/L)*sin(2*pi*j*t/L);
I3 = int(f3,t,0,L)

when i am integrating the function f3(t) from limits 0 to L, the code is not giving me the output in the form of some expression. The output i am getting is in the form of int(f3,t,0,L), which I dont want. I want output in the form of expression (which will contain t L rho m L n T k G v).

Please help me with this!! Any help will be appreciated..

[Walter Roberson]

It is not obvious that a closed form integral for that exists.

[aakash dewangan]

Hello Walter Roberson,

Thanks for reply. But in my question when i am providing the values of all the parameters (L rho m L n T k G v) then i am getting the answer in the form of some numerical value. Therefore, I am expecting the expression when values of parameters are not given. Could you please clarify this?

Thanks in advance

[Walter Roberson]

Perhaps you got lucky, or perhaps you made a mistake.

In the below, I substitute in numeric values for everything other than t, but matlab is not able to find the integral

syms  t real
Pi = sym(pi);

rho = sym(randi([-99 99]))/10;
m = sym(randi([-99 99]))/10;
n = sym(randi([-99 99]))/10;
T = sym(randi([-99 99]))/10;
k = sym(randi([-99 99]))/10;
G = sym(randi([-99 99]))/10;
v = sym(randi([-99 99]))/10;
L = sym(randi(99))/10;

N(t) = (T*L/2)*(n*Pi/L)^2+k*sin(n*Pi*v*t/L);
D(t) = (rho*L/2+m*(sin(n*Pi*v*t/L))^2);
alpha(t) = N/D;
i = 6; j = 1;
f3(t) = alpha(t)*cos(2*Pi*i*t/L)*sin(2*Pi*j*t/L);
I3 = int(f3, t, [0,L])

I3 = 

[aakash dewangan]

Walter Roberson, thanks again for your reply and help.

Yes, you are right. I got integration output for a few combinations of parameter values (not for all values). Thanks for help.

"Do you have any suggestion for me to do this (integration) using any other approach / method / technique??"

Thanks,

[Walter Roberson]

I have no suggestions.

There just might be a change of variables available to make sin(n*pi*v*t/L) linear.

Is n*v known to be integer? If so then that would make a difference in the integration, as sin(n*pi*v*t/L) at t=L would be sin(n*pi*v) and if n*v were integer that would be 0 .

Answer 1

[Jan]

syms  t L rho m L n T  k  G  v
N(t) = (T*L/2)*(n*pi/L)^2+k*sin(n*pi*v*t/L);
D(t) = (rho*L/2+m*(sin(n*pi*v*t/L))^2);
alpha(t) = N/D;
i = 6; j = 1;
f3(t) = alpha(t)*cos(2*pi*i*t/L)*sin(2*pi*j*t/L);
I3 = int(f3, t, [0,L])

I3 = 

As Walter has said already: There is no closed form solution of this integral. You can simplify it only, if you provide specific values for the parameters.