For {x^2} ne npi + 1, n in N (the set of natu | vedclass

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(D) Let $I = \int x \sqrt{\frac{2 \sin(x^2-1) - 2 \sin(x^2-1) \cos(x^2-1)}{2 \sin(x^2-1) + 2 \sin(x^2-1) \cos(x^2-1)}} dx$ $I = \int x \sqrt{\frac{2 \

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${\log _e}\left| {\sec \left( {\frac{{{x^2} - 1}}{2}} \right)} \right| + c$

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(D) Let $I = \int x \sqrt{\frac{2 \sin(x^2-1) - 2 \sin(x^2-1) \cos(x^2-1)}{2 \sin(x^2-1) + 2 \sin(x^2-1) \cos(x^2-1)}} dx$ $I = \int x \sqrt{\frac{2 \sin(x^2-1) [1 - \cos(x^2-1)]}{2 \sin(x^2-1) [1 + \cos(x^2-1)]}} dx$ $I = \int x \sqrt{\frac{2 \sin^2(\frac{x^2-1}{2})}{2 \cos^2(\frac{x^2-1}{2})}} dx$ $I = \int x 	an(\frac{x^2-1}{2}) dx$ Let $t = \frac{x^2-1}{2}$,then $dt = x dx$ $I = \int 	an(t) dt = \ln|\sec(t)| + c$ $I = \ln|\sec(\frac{x^2-1}{2})| + c$

For ${x^2} \ne n\pi + 1, n \in N$ (the set of natural numbers),the integral $\int {x\sqrt {\frac{{2\sin \left( {{x^2} - 1} \right) - \sin 2\left( {{x^2} - 1} \right)}}{{2\sin \left( {{x^2} - 1} \right) + \sin 2\left( {{x^2} - 1} \right)}}} } dx$ is

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