$$f(x) =\begin{vmatrix} x^3 & \cos^2x & 2x^4 \\ \tan^3x & 1 & \sec2x \\ \sin^3x & x^4 & 5 \\ \end{vmatrix} $$
Then $\int^{\pi/2}_{-\pi/2} \det(f(x))\,dx$ is equal to ?
The way I thought of solving it was by making use of the fact some of these are odd functions and Hence their integration along the given intervals would give a result of zero , but since some of them are even functions i cant directly remove them . I thought of opening up the determinant of the 3x3 matrix ,but this would result in even more complicated functions .
This question was in a competitive exam where on average 3 mins can be given to per question ,so is there an approach do solve it In a more decent way ?
The answer given is $0$.
Notice that $f(x)=-f(-x)$ by linearity of the determinant. So $f$ is an odd function being integrated on a symmetric interval around 0...