Let n ≥ 2 be a natural number and \(0 < \theta < \frac{\pi }{2}.{\rm{\;Then\;}}\smallint \left( {\frac{{{{\left( {{\rm{si}}{{\rm{n}}^n}\theta - {\rm{sin}}\theta } \right)}^{\frac{1}{n}}}{\rm{cos}}\theta }}{{{\rm{si}}{{\rm{n}}^{n + 1}}\theta }}} \right)d\theta\) is equal to:(Where C is a constant of integration)

• A. \(\frac{{\rm{n}}}{{{{\rm{n}}^2} - 1}}{\left( {1 - \frac{1}{{{\rm{si}}{{\rm{n}}^{{\rm{n}} - 1}}\theta }}} \right)^{\frac{{{\rm{n}} + 1}}{{\rm{n}}}}} + {\rm{C}}\)
• B. \(\frac{{\rm{n}}}{{{{\rm{n}}^2} + 1}}{\left( {1 - \frac{1}{{{\rm{si}}{{\rm{n}}^{{\rm{n}} - 1}}{\rm{\theta }}}}} \right)^{\frac{{{\rm{n}} + 1}}{{\rm{n}}}}} + {\rm{C}}\)
• C. \(\frac{{\rm{n}}}{{{{\rm{n}}^2} - 1}}{\left( {1 + \frac{1}{{{\rm{si}}{{\rm{n}}^{{\rm{n}} - 1}}{\rm{\theta }}}}} \right)^{\frac{{{\rm{n}} + 1}}{{\rm{n}}}}} + {\rm{C}}\)
• D. \(\frac{{\rm{n}}}{{{{\rm{n}}^2} - 1}}{\left( {1 - \frac{1}{{{\rm{si}}{{\rm{n}}^{{\rm{n}} + 1}}\theta }}} \right)^{\frac{{{\rm{n}} + 1}}{{\rm{n}}}}} + {\rm{C}}\)