Explanation:

To find the value of the Jacobian for the integral cos(xy) dxdy using the transformation x = u and y = v, we need to perform the change of variables.

Change of Variables:
Let's first determine the Jacobian of the transformation, which is given by the determinant of the matrix of partial derivatives of the transformation.

J = |∂x/∂u ∂x/∂v|
|∂y/∂u ∂y/∂v|

Since x = u and y = v, the partial derivatives are as follows:

∂x/∂u = 1
∂x/∂v = 0
∂y/∂u = 0
∂y/∂v = 1

Therefore, the Jacobian J is:

J = |1 0|
|0 1|

Integral Transformation:
Now, let's perform the change of variables in the integral:

∫∫ cos(xy) dxdy

Using the transformation x = u and y = v, we have:

∫∫ cos(uv) |J| dudv

Since the Jacobian J is a constant, we can take it outside the integral:

J = |1 0|
|0 1|

∫∫ cos(uv) dudv

Solving the Integral:
To solve the integral, we can integrate with respect to u first, treating v as a constant:

∫ cos(uv) du = (1/v) sin(uv) + C1

Now, we integrate this result with respect to v:

∫ [(1/v) sin(uv) + C1] dv = (1/v^2) (1/v) sin(uv) + C1v + C2

Therefore, the final result of the integral is:

∫∫ cos(xy) dxdy = (1/v^2) (1/v) sin(uv) + C1v + C2

Summary:
- The Jacobian of the transformation x = u and y = v is J = |1 0| |0 1|.
- The integral cos(xy) dxdy can be transformed by substituting x = u and y = v.
- The integral becomes ∫∫ cos(uv) dudv.
- The integral can be solved by integrating with respect to u and v separately.
- The final result of the integral is (1/v^2) (1/v) sin(uv) + C1v + C2.