- #1
DryRun
Gold Member
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Homework Statement
http://s2.ipicture.ru/uploads/20120109/dT4m6rNG.jpg
The attempt at a solution
[tex]x=\frac{u}{1+v}[/tex] and [tex]y=\frac{uv}{1+v}[/tex]
Transforming the integrand: [tex]\frac{x+y}{x^2}e^{x+y}=\frac{(1+v)^2 e^u}{u}[/tex]
[tex]dxdy=J.dudv[/tex]
[tex]J=\frac{v(1+v)^2 +1+uv}{(1+v)^3}[/tex]
The double integral becomes: [tex]\int\int \frac{e^u[v(1+v)^2+1+uv]}{u(1+v)}.dudv[/tex]
This is the part where I'm having some trouble as i broke the large integrand into smaller ones:
[tex]\int\frac{e^u[v(1+v)^2+1+uv]}{u(1+v)}.du=\int\frac{e^u v (1+v)}{u}.du+\int\frac{e^u}{u(1+v)}.du+\int\frac{e^u v}{(1+v)}.du[/tex]
The integral of [itex]\frac{e^u}{u}[/itex] is what's blocking my progress through this problem. I did an online search and it appears that it can't be solved unless i use some special function that mathematicians invented called exponential integral. So, i think i might have made an error somewhere and might have inadvertently overstepped into unknown territory. I haven't started working on the limits yet, as i wouldn't be able to integrate at this point.
http://s2.ipicture.ru/uploads/20120109/dT4m6rNG.jpg
The attempt at a solution
[tex]x=\frac{u}{1+v}[/tex] and [tex]y=\frac{uv}{1+v}[/tex]
Transforming the integrand: [tex]\frac{x+y}{x^2}e^{x+y}=\frac{(1+v)^2 e^u}{u}[/tex]
[tex]dxdy=J.dudv[/tex]
[tex]J=\frac{v(1+v)^2 +1+uv}{(1+v)^3}[/tex]
The double integral becomes: [tex]\int\int \frac{e^u[v(1+v)^2+1+uv]}{u(1+v)}.dudv[/tex]
This is the part where I'm having some trouble as i broke the large integrand into smaller ones:
[tex]\int\frac{e^u[v(1+v)^2+1+uv]}{u(1+v)}.du=\int\frac{e^u v (1+v)}{u}.du+\int\frac{e^u}{u(1+v)}.du+\int\frac{e^u v}{(1+v)}.du[/tex]
The integral of [itex]\frac{e^u}{u}[/itex] is what's blocking my progress through this problem. I did an online search and it appears that it can't be solved unless i use some special function that mathematicians invented called exponential integral. So, i think i might have made an error somewhere and might have inadvertently overstepped into unknown territory. I haven't started working on the limits yet, as i wouldn't be able to integrate at this point.
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