Let \( x^{2} \neq n \pi-1, n \in N \), then \( \mathrm{P}^{11:} \) ...

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Let \( x^{2} \neq n \pi-1, n \in N \), then
\( \mathrm{P}^{11:} \) \( \int x \cdot \sqrt{\frac{2 \sin \left(x^{2}+1\right)-\sin 2\left(x^{2}+1\right)}{2 \sin \left(x^{2}+1\right)+\sin 2\left(x^{2}+1\right)}} d x \) is equal to:
(1) \( \ln \left|\frac{1}{2} \sec \left(x^{2}+1\right)\right|+c \)
(2) \( \frac{1}{2} \ln \left|\sec \left(x^{2}+1\right)\right|+c \)
(3) \( \ln \left|\sec \left(\frac{x^{2}+1}{2}\right)\right|+c \)
(4) \( \frac{1}{2} \ln \left|\frac{2}{\sec \left(x^{2}+1\right)}\right|+c \) \( \equiv \square \)
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