For x2 ≠ nπ + 1, n ∈ N (the set of natural numbers), the integral \(\smallint {\rm{x}}\sqrt {\frac{{2{\rm{sin}}\left( {{{\rm{x}}^2} - 1} \right) - {\rm{sin}}2\left( {{{\rm{x}}^2} - 1} \right)}}{{2{\rm{sin}}\left( {{{\rm{x}}^2} - 1} \right) + {\rm{sin}}2\left( {{{\rm{x}}^2} - 1} \right)}}} {\rm{dx}}\) is equal to:(where c is a constant of integration)

• A. \({\rm{lo}}{{\rm{g}}_{\rm{e}}}\left| {\frac{1}{2}{\rm{se}}{{\rm{c}}^2}\left( {{{\rm{x}}^2} - 1} \right)} \right| + {\rm{c}}\)
• B. \(\frac{1}{2}{\rm{lo}}{{\rm{g}}_{\rm{e}}}\left| {{\rm{sec}}\left( {{{\rm{x}}^2} - 1} \right)} \right| + {\rm{c}}\)
• C. \(\frac{1}{2}{\rm{lo}}{{\rm{g}}_{\rm{e}}}\left| {{\rm{se}}{{\rm{c}}^2}\left( {\frac{{{{\rm{x}}^2} - 1}}{2}} \right)} \right| + {\rm{c}}\)
• D. \({\rm{lo}}{{\rm{g}}_{\rm{e}}}\left| {{\rm{sec}}\left( {\frac{{{{\rm{x}}^2} - 1}}{2}} \right)} \right| + {\rm{c}}\)