Given, f(x)=| 0 x^2-sin x cos x-2 sin x-x^2 ...

Given, $f(x)=\left|\begin{array}{ccc}0 & x^{2}-\sin x & \cos x-2 \\ \sin x-x^{2} & 0 & 1-2 x \\ 2-\cos x & 2 x-1 & 0\end{array}\right|$, then $\int f(x) d x$ is equal to (a) $\frac{x^{3}}{3}-x^{2} \sin x+\sin 2 x+C$ (b) $\frac{x^{3}}{3}-x^{2} \sin x-\cos 2 x+C$ (c) $\frac{x^{3}}{3}-x^{2} \cos x-\cos 2 x+C$ (d) None of the above

Given, $f(x)=\left|\begin{array}{ccc}0 & x^{2}-\sin x & \cos x-2 \\ \sin x-x^{2} & 0 & 1-2 x \\ 2-\cos x & 2 x-1 & 0\end{array}\right|$, then
$\int f(x) d x$ is equal to
(a) $\frac{x^{3}}{3}-x^{2} \sin x+\sin 2 x+C$
(b) $\frac{x^{3}}{3}-x^{2} \sin x-\cos 2 x+C$
(c) $\frac{x^{3}}{3}-x^{2} \cos x-\cos 2 x+C$
(d) None of the above

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Key Concepts

Determinants

A determinant is a scalar value that is computed from a square matrix and is used to characterize certain properties of the matrix, such as whether it is invertible and how it scales volume. Determinants are calculated through specific formulas that involve sums of products of matrix entries, and they play a critical role in linear algebra.

Skew?Symmetric Matrices

A skew?symmetric matrix is one in which the transpose of the matrix is equal to the negative of the matrix. This means that for every entry above the main diagonal, there is a corresponding entry below the diagonal that is its negative. This property has profound implications in many areas of mathematics, including the fact that the determinant of any odd-order skew?symmetric matrix is always zero.

Integration and Antiderivatives

Integration is the process of finding a function whose derivative is a given function, known as the antiderivative. When the function to be integrated is identically zero, the antiderivative is a constant function. This concept is fundamental in calculus and is used to solve a wide variety of problems involving accumulation and area calculations.

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