For x2 $ \ne $ n$\pi $ + 1, n $ \in $ N (the set of natural numbers), the integral $\int {x\sqrt {{{2\sin ({x^2} - 1) - \sin 2({x^2} - 1)} \over {2\sin ({x^2} - 1) + \sin 2({x^2} - 1)}}} dx} $ is equal to : (where c is a constant of integration)

Question

For x² $ \ne $ n$\pi $ + 1, n $ \in $ N (the set of natural numbers), the integral

$\int {x\sqrt {{{2\sin ({x^2} - 1) - \sin 2({x^2} - 1)} \over {2\sin ({x^2} - 1) + \sin 2({x^2} - 1)}}} dx} $ is equal to :

(where c is a constant of integration)

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

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

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

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

Solution

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