Solve a double integral expression (avoid no-convergence related error)

Questions
1

I am having problem solving a double integral in which the error is regarding that the integral is not converging. I have tried to solve on paper and also to find identity related to the expression but was unable to do so. I am attaching the notebook with the expression and output corresponding to it. It would be helpful for me if I can get some guidance for the same.

Thank you.

First.nb (5.4 KB)

2

Your notebook does not open (on my system). Error message: file corrupted.

3

Sorry, I am attaching the document again. The first cell is the main expression and expression followed after ArcSin should be in square brackets, also the limit for second integration is 0 to Infinity. As I am trying to change my file in a text editor in Linux rather than windows where I first created the file, the file got corrupted.

new.nb (69.8 KB)

4

Hi Ratnam,

There are syntax errors in the code e.g.

* [\[Pi] + 2 ArcSin ((2*(y^2 - (b - h)^2)/y^2) - 1)])

Fix the errors, they are highlighted in the notebook in red and in the error messages.

Syntax::sntxf: “(y/4ab)" cannot be followed by
"[[Pi]+2ArcSin((2(y^2-(b-h)^2)/y^2)-1)]”.

5

Sorry for the trouble because of the file. I am attaching the perfect document it is comprised of expression and also the error.

first.nb (4.8 KB)

6

I think there is no solution.

The first (inner) integral is found to be

In[4]:= Integrate[x*e^(-x^2), {x, 0, y*z}] // FullSimplify

Out[4]= -((-1 + e^(-y^2 z^2))/(2 Log[e]))

which is essentially a constant and an exponential Function, a + e[x] say. Insert this in the integrand for the “outer” integral and multiply things out:

In[6]:= (a + e[x])*(y/4*a*b)*(\[Pi] +2 ArcSin[(2*(y^2 - (b - h)^2)/y^2) - 1]) // Expand

Out[6]= 1/4 a^2 b \[Pi] y - 1/2 a^2 b y ArcSin[1 - (2 (-(b - h)^2 + y^2))/y^2] + 1/4 a b \[Pi] y e[x] -  
1/2 a y ArcSin[1 - (2 (-(b - h)^2 + y^2))/y^2] e[x]

The first term is a linear function of y, and the corresponding integral, which is essentially y^2 in the bounds 0 and infinity of course tends to infinity or doesn’t converge:

In[7]:= Integrate[1/4 a^2 b \[Pi] y, {y, 0, Infinity}]

During evaluation of In[7]:= Integrate::idiv: Integral of y does not converge on {0,\[Infinity]}. >>