Integrating a determinant

Question

This:

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.

Answer 1

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.

Answer 2

$$f(x) = (x^2 - \sin x)(1-2x)^2 + (2x-1)(x^2 - \sin x )(1-2x) = 0 $$ Then $$\int f(x) = C$$ which is a constant.