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For values of x satisfying x^2 \ne n\pi + 1, where n \in \mathbb{N}, evaluate the integral \[\int x\,\frac{\sqrt{2\sin(x^2-1)-\sin\big(2(x^2-1)\big)}}{\sqrt{2\sin(x^2-1)+\sin\big(2(x^2-1)\big)}}\,dx\]

1. \(\log_e\left|\tfrac12\sec^2(x^2-1)\right|+c\)
2. \(\tfrac12\log_e\left|\sec(x^2-1)\right|+c\)
3. \(\tfrac12\log_e\left|\sec^2\!\left(\frac{x^2-1}{2}\right)\right|+c\)
4. \(\log_e\left|\sec\!\left(\frac{x^2-1}{2}\right)\right|+c\)

Correct answer: \(\tfrac12\log_e\left|\sec^2\!\left(\frac{x^2-1}{2}\right)\right|+c\)

Solution

The correct option is right because the integral simplifies to a form involving the secant function, and the factor of 1/2 arises from the substitution used in the integration process, specifically relating to the derivative of the argument of the secant function.

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