$\int_0^{\pi /2} {{{\sin }^{2m}}} xdx = \frac{{(2m - 1)}}{{2m}}.\frac{{(2m - 3)}}{{(2m - 2)}}.....\frac{3}{4}.\frac{1}{2}.\frac{\pi }{2}$
$ = \frac{{2m.(2m - 1)(2m - 2)....3.2.1.\frac{\pi }{2}}}{{{{[2m.(2m - 2)(2m - 4).....4.2]}^2}}}$
Multiplying the numerator and the denominator by $2m(2m - 2)....4.2$
$ = \frac{{(2m)!}}{{{{[{2^m}.m(m - 1)(m - 2).....2.1]}^2}}}\frac{\pi }{2}$
$ = \frac{{(2m)!}}{{{{({2^m}.m!)}^2}}}\frac{\pi }{2}$ .
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.