Integrating a determinant

determinantintegration

This:

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Am i allowed to apply the same rules of diiferentiation i.e. first differentiating any row/column and keeping the rest same to integration(integrating any row/colum first and keeping rest same and proceeding like this for the other row/column)?
If not how to tackle this?

I can see it's a skew symmetric matrix and the determinant is zero but it does not seem to help.

Best Answer

If a function is $0$, then its only antiderivatives are constant functions. It doesn't matter that the expression of the function is using determinants or trigonometric expressions.

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