Q # 9 please resolve this
1 answer:
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Consider substituting 
, so that 
. Then
Then by similar substitutions, you can easily find that you end up with
Of course, this all assumes that the integrand is continuous over the domain of integration, which would require that
are chosen such that 
for any ![(x,y)\in[a,b]^2](https://tex.z-dn.net/?f=%28x%2Cy%29%5Cin%5Ba%2Cb%5D%5E2)
. If in particular 
, then we can write
and you can combine the logarithms if you like as
![=\ln\sqrt[a]{\dfrac{4+ab}{4+a^2}}+\ln\sqrt[b]{\dfrac{4+ab}{4+b^2}}](https://tex.z-dn.net/?f=%3D%5Cln%5Csqrt%5Ba%5D%7B%5Cdfrac%7B4%2Bab%7D%7B4%2Ba%5E2%7D%7D%2B%5Cln%5Csqrt%5Bb%5D%7B%5Cdfrac%7B4%2Bab%7D%7B4%2Bb%5E2%7D%7D)
![=\ln\left(\sqrt[a]{\dfrac{4+ab}{4+a^2}}\sqrt[b]{\dfrac{4+ab}{4+b^2}}\right)](https://tex.z-dn.net/?f=%3D%5Cln%5Cleft%28%5Csqrt%5Ba%5D%7B%5Cdfrac%7B4%2Bab%7D%7B4%2Ba%5E2%7D%7D%5Csqrt%5Bb%5D%7B%5Cdfrac%7B4%2Bab%7D%7B4%2Bb%5E2%7D%7D%5Cright%29)
Then by similar substitutions, you can easily find that you end up with
Of course, this all assumes that the integrand is continuous over the domain of integration, which would require that
and you can combine the logarithms if you like as