explicit integral could not be found problem

Hello all,

I want to solve for definite integral. It is little complicated. I was unable to write it over here that is why I have attached pdf file along with this. It just contains 3 equations.

The matlab code for this is as follows:

clear all;

pd=0.015; dr=0.0614;

phi1=acosd(pd/(2*dr));

epsi=0.5*(1-pd/(2*dr));

syms theta;

f=(1/(2*pi))*(1-(0.5/epsi)*(1-cos(theta)))^(1.5)*cos(theta);

answer=int(f,theta,-phi1,phi1);

When I run the code I am getting following output:

Warning: Explicit integral could not be found. 
>> answer

answer =

int((5734161139222659*cos(theta)*((614*cos(theta))/539 - 75/539)^(3/2))/36028797018963968, theta = -5839466350469371/70368744177664..5839466350469371/70368744177664)

I have no clue how to debug this. Can somebody please help me how to solve this integral problem using MATLAB. For integral please refer to pdf file.

Thanks in advance

Nikhil

[Nikhil]

integral equation is in the pdf file

[Nikhil]

in case if der is problem with pdf file, I am writing equations over here ε= 1/2(1-pd/(2*dr))

θ_1=cos^(-1)(pd/(2*dr))

Jr=1/((2*pi)) * integration -θ_1 to θ_1 of f(theta)

where f(θ)=[1-(1/(2*ε) )*(1-cos(θ))]^1.5 ×cosθ Where, pd=0.015, dr=0.0614 I want to solve for Jr

[Walter Roberson]

Why did you use acosd instead of acos in defining phi1 ?

[Nikhil]

Hey, bcoz I wanted phi1 to be in degrees.

[Walter Roberson]

No you don't. You use phi1 as your bounds on the integral, where it becomes your theta variable, but you take cos() of theta not cosd(theta), so theta must be in radians so phi1 should be in radians.

Answer 1

[Walter Roberson]

syms theta

pd = 15/1000;       %symbolics are easier to get right when rationals are used
dr = 614/10000;

epsilon = 1/2*(1-pd/(2*dr));
theta1 = acos(pd/(2*dr));

T = (1-(1-cos(theta))/(2*epsilon))^(3/2)*cos(theta);

f = simplify( 1/(2*pi) * int(T, theta, -theta1, theta1) );

Using Maple, the result I get is

 (1/127414210)*(2273226*3377^(1/2)*elliptice((2/77)*sin((1/2)*acos(75/614))*3377^(1/2), (7/614)*3377^(1/2))-1320813*3377^(1/2)*ellipticf((2/77)*sin((1/2)*acos(75/614))*3377^(1/2), (7/614)*3377^(1/2))+(1228*sin((3/2)*acos(75/614))-1828*sin((1/2)*acos(75/614)))*(617562121+636557746*cos(2*acos(75/614)))^(1/2)+2273226*(-(440271/757742)*elliptick((7/614)*3377^(1/2))+elliptice((7/614)*3377^(1/2)))*3377^(1/2))/pi

Which is approximately

0.218945210999261

[Nikhil]

Hey, thanks a lot for your analysis. But I want to solve this problem using Matlab. Because this code is a part of larger code. Hence I want to get this logic run using Matlab. Can you tell me how to solve this using Matlab? Thanks in advance

[Alan Weiss]

You can use the double command to get the answer in numeric form:

 pd = sym('15/1000');
 dr = sym('614/10000');
 epsilon = 1/2*(1-pd/(2*dr));
 theta1 = acos(pd/(2*dr));
 T = (1-(1-cos(theta))/(2*epsilon))^(3/2)*cos(theta);
 f = simplify( 1/(2*pi) * int(T, theta, -theta1, theta1) ); % so far just like Walter's answer
 f2 = double(f)

 f2 =

    0.2189

All I mean is you can use the answer for subsequent numeric calculations, even though it is not explicit like the Maple answer.

Alan Weiss

MATLAB mathematical toolbox documentation

[Walter Roberson]

Ah, right. Should probably sym() the 1/2 and the 3/2 as well, and maybe the pi too.

[Walter Roberson]

Nikhil, I do not have the Symbolic Toolbox, so I cannot report the exact form of the answer that MATLAB would give. Sometimes I am able to figure out the MuPAD commands from the documentation, but I cannot always predict the MuPAD output.

[Nikhil]

Hey Alan and Walter, Thanks a lot for your help. Alan, when I run your code, i.e. when I run the code mentioned below, then I am getting an following error:

syms theta;
pd = sym('15/1000');
 dr = sym('614/10000');
 epsilon = 1/2*(1-pd/(2*dr));
 theta1 = acos(pd/(2*dr));
 T = (1-(1-cos(theta))/(2*epsilon))^(3/2)*cos(theta);
 f = simplify( 1/(2*pi) * int(T, theta, -theta1, theta1) ); % so far just like Walter's answer
 f2 = double(f)

Error: Warning: Explicit integral could not be found. >> f

f =

(47389761481179*11^(1/2)*int(cos(theta)*(614*cos(theta) - 75)^(3/2), theta = -acos(75/614)..acos(75/614)))/12357877377504641024

??? Error using ==> mupadmex Error in MuPAD command: DOUBLE cannot convert the input expression into a double array.

If the input expression contains a symbolic variable, use the VPA function instead.

Error in ==> sym.sym>sym.double at 936 Xstr = mupadmex('symobj::double', S.s, 0);

So in all I am getting these two errors, after running this code. Do I need to have certain tool box to solve for these, because I am just using student version? Thanks in advance,

Nikhil