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JEE MATHEMATICS MCQ: INDEFINITE INTEGRATION

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Let x2≠n  π−1,   n∈N{x^2} \ne n\;\pi - 1,\;\,n \in N. The indefinite integral of
x  2  sin⁡  (x2+  1)  −  sin⁡  2(x2+  1)2  sin⁡  (x2+  1)  +  sin⁡  2(x2+  1)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)}}}

is but for an arbitrary constant
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Correct option: B
log⁡  ∣ 12  sec⁡  (x2+12) ∣\log \,\,\left| {\,\frac{1}{2}\,\,\sec \,\,\left( {\frac{{{x^2} + 1}}{2}} \right)\,} \right|
Option B (\log \,\left| {\\,\frac{1}{2}\\,\sec \\,\left( {\frac{{{{{x^2} + 1}}}}{2}} \right)\} \right|\log \,\left| {\\,\frac{1}{2}\\,\sec \\,\left( {\frac{{{{{x^2} + 1}}}}{2}} \right)\} \right|) is correct because it is the result of the indefinite integral after simplification and substitution.

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