By inspecting the integrand, the "obvious" choice for substitution would be
<em>u</em> = <em>y</em> + <em>x</em>
<em>v</em> = <em>y</em> - <em>x</em>
<em />
Solving for <em>x</em> and <em>y</em>, we would have
<em>x</em> = (<em>u</em> - <em>v</em>)/2
<em>y</em> = (<em>u</em> + <em>v</em>)/2
in which case the Jacobian and its determinant are
![J=\begin{bmatrix}x_u&x_v\\y_u&y_v\end{bmatrix}=\dfrac12\begin{bmatrix}1&-1\\1&1\end{bmatrix}\implies|\det J|=\left|\dfrac12\right|=\dfrac12](https://tex.z-dn.net/?f=J%3D%5Cbegin%7Bbmatrix%7Dx_u%26x_v%5C%5Cy_u%26y_v%5Cend%7Bbmatrix%7D%3D%5Cdfrac12%5Cbegin%7Bbmatrix%7D1%26-1%5C%5C1%261%5Cend%7Bbmatrix%7D%5Cimplies%7C%5Cdet%20J%7C%3D%5Cleft%7C%5Cdfrac12%5Cright%7C%3D%5Cdfrac12)
The trapezoid <em>R</em> has two of its edges on the lines <em>x</em> + <em>y</em> = 8 and <em>x</em> + <em>y</em> = 9, so right away, we have 8 ≤ <em>u</em> ≤ 9.
Then for <em>v</em>, we observe that when <em>x</em> = 0 (the lowest edge of <em>R</em>), <em>v</em> = <em>y</em> ; similarly, when <em>y</em> = 0 (the leftmost edge of <em>R</em>), <em>v</em> = -<em>x</em>. So
-<em>x</em> ≤ <em>v</em> ≤ <em>y</em>
-(<em>u</em> - <em>v</em>)/2 ≤ <em>v</em> ≤ (<em>u</em> + <em>v</em>)/2
-<em>u</em> + <em>v</em> ≤ 2<em>v</em> ≤ <em>u</em> + <em>v</em>
-<em>u</em> ≤ <em>v</em> ≤ <em>u</em>
<em />
So, the integral becomes
![\displaystyle\iint_R5\cos\left(7\frac{y-x}{y+x}\right)\,\mathrm dA=\int_8^9\int_{-u}^u\frac52\cos\left(\frac{7v}u\right)\,\mathrm dv\,\mathrm du](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Ciint_R5%5Ccos%5Cleft%287%5Cfrac%7By-x%7D%7By%2Bx%7D%5Cright%29%5C%2C%5Cmathrm%20dA%3D%5Cint_8%5E9%5Cint_%7B-u%7D%5Eu%5Cfrac52%5Ccos%5Cleft%28%5Cfrac%7B7v%7Du%5Cright%29%5C%2C%5Cmathrm%20dv%5C%2C%5Cmathrm%20du)
![=\displaystyle\frac52\int_8^9\frac u7(\sin7-\sin(-7))\,\mathrm du](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle%5Cfrac52%5Cint_8%5E9%5Cfrac%20u7%28%5Csin7-%5Csin%28-7%29%29%5C%2C%5Cmathrm%20du)
![=\displaystyle\frac57\sin7\int_8^9u\,\mathrm du](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle%5Cfrac57%5Csin7%5Cint_8%5E9u%5C%2C%5Cmathrm%20du)
![=\displaystyle\frac5{14}\sin7(9^2-8^2)=\boxed{\frac{85}{14}\sin7}](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle%5Cfrac5%7B14%7D%5Csin7%289%5E2-8%5E2%29%3D%5Cboxed%7B%5Cfrac%7B85%7D%7B14%7D%5Csin7%7D)