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If $f (x) =$ $|\begin{array}{ccc} \sin x + \sin 2 x + \sin 3 x & \sin 2 x & \sin 3 x \\ 3 + 4 \sin x & 3 & 4 \sin x \\ 1 + \sin x & \sin x & 1 \end{array}|$ then the value of $\int_{0}^{π / 2} f (x) d x$ is

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Correct option is D

By applying the operation C1→C1−C2−C3,f(x) can be written as f(x)=sin⁡xsin⁡2xsin⁡3x034sin⁡x0sin⁡x1=sin⁡x3−4sin2⁡x=sin⁡3xSo ∫0π/2 f(x)dx=−cos⁡3x30π/2=13.

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If f(x)= sin⁡x+sin⁡2x+sin⁡3xsin⁡2xsin⁡3x3+4sin⁡x34sin⁡x1+sin⁡xsin⁡x1 then the value of ∫0π/2 f(x)dx is

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If $f (x) =$ $|\begin{array}{ccc} \sin x + \sin 2 x + \sin 3 x & \sin 2 x & \sin 3 x \\ 3 + 4 \sin x & 3 & 4 \sin x \\ 1 + \sin x & \sin x & 1 \end{array}|$ then the value of $\int_{0}^{π / 2} f (x) d x$ is