Solution of given Integration

Please help to solve the integration given below:

where K1, K2, A, B and C are constants. The dummy variable is x.

[madhan ravi]

upload your code that you tried and the datas , gamma function too??

[Shashibhushan Sharma]

Sorry sir, this integration is solved by matlab code. There is no problem to solve it by matlab code, but I want to solve it in closed form. Can you help to solve it in closed form?

[Walter Roberson]

No. It is not possible to do a closed form integration of arbitrary unknown functions multiplied by something .

[Shashibhushan Sharma]

is a lower incomplte gamma function. is an exponential function.

[Walter Roberson]

igamma for symbolic upper incomplete gamma function . The description shows how to calculate lower incomplete . Watch out for the order of parameters .

[John D'Errico]

Do you know a closed form solution must exist for general A,B,C,K1,K2?

[Shashibhushan Sharma]

let A=5, B=4,C=6, K1=2, K2=30;

[Walter Roberson]

I am finding two different definitions for the lower incomplete gamma function. The one given in the igamma() definition at https://www.mathworks.com/help/symbolic/igamma.html#bt6_p8p-1 corresponds to int(t^(nu-1)*exp(-t),t=0..z) but the one given at https://www.mathworks.com/help/matlab/ref/gammainc.html#bvghju3-1 is 1/gamma(a)* int(t^(a-1)*exp(-t),t=0..z) . I do not know if the difference between calling the parameter "nu" or "a" is significant; I suppose it is possible that there are two different conventions and that hypothetically there might be some linear scaling going on . In any case, we need to know which version you want, the version that is reduced by gamma() of the first argument or not ?

[Walter Roberson]

It looks to me as if no closed form solution exists for those particular constants. It looks like it comes out as

int(-exp(-6/x)*(exp(-4*x)*(1+4*x+8*x^2+32/3*x^3+32/3*x^4)-1)/x^2,x = 2 .. 30)

or a constant multiple of that.

[Shashibhushan Sharma]

I think, it is not possible to solve in closed form.

Answer 1

[madhan ravi]

gamma = @(A,B) A .* B .* cos( A.*B ) ; % an example how to proceed 
A = 4 ;
B = 6 ;
C = 10 ; 
fun = @(x) ( gamma( A , B .* x ) .* exp( C ./ x) ) ./ x.^2 ;
K1 = 8 ;
K2 = 13 ; 
Result = integral( fun , K1 , K2 )

[Shashibhushan Sharma]

Thnak for trying to give answer of my question.

But Function should be written as:

fun = @(x) ( gamma(A).*gammainc(B .* x, A ) .* exp( C ./ x) ) ./ x.^2 ;

But I want a closed form solution of this integration.