If , then is equal to

Engineering

Mathematics

Special Type of Square Matrices

Properties of Determinant

Modulus Function

Question

If $f (x) = |\begin{array}{ccc} 0 & x^{2} - \sin x & \cos x - 2 \\ \sin x - x^{2} & 0 & 1 - 2 x \\ 2 - \cos x & 2 x - 1 & 0 \end{array}|$ , then $\int f (x) dx$ is equal to

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f (x) = |\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}|

= -– (x² – sin⁡x) (–(2 – cos⁡x)(1 – 2x)) + (cos⁡x – 2) (sin⁡x – x²(2x – 1))

= (x² – sin⁡x) (2 – cos⁡x) (1 – 2x) + (2 – cos⁡x)(sin⁡x – x²)(1 – 2x)

= (x² – sin⁡x) (2 – cos⁡x) (1 – 2x) – (2 – cos⁡x)(x² – sin⁡x) (1 – 2x) = 0

∴ \int f (x) dx = \int odx =

constant.