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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.

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  • $\begingroup$ @ Arthur It seems too much work . I am looking for some easy way around . I tried by row transformations but it got more complicated. $\endgroup$
    – Starboy
    Apr 1, 2019 at 15:41
  • $\begingroup$ Your function $f(x)$ is the determinant, so if (as you note) the determinant vanishes then $f(x)=0$ and so too does the integral (up to the usual additive constant). I guess I’ll also point out that every skew-symmetric matrix has zero determinant. $\endgroup$ Apr 1, 2019 at 15:44
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    $\begingroup$ @Semiclassical That's only true for odd-dimensional matrices. $2n\times 2n$ matrices can be skew-symmetric and invertible at the same time, like $\left[\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}\right]$. $\endgroup$
    – Arthur
    Apr 1, 2019 at 17:57

2 Answers 2

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

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