If the integral 525 ∫ _ 0 ^ (π)/(2) 2xcos(11)/(2)x(1 +...

Let,

$I = \mathop{∫}_{0}^{\dfrac{\pi}{2}}\sin 2xcos\dfrac{11}{2}x\left(1 + cos\dfrac{5}{2}x\right)\dfrac{1}{2}dx$

Let $\cos x = {t}^{2} ⇒ sinxdx = - 2tdt$

$∴ I = - 4\mathop{∫}_{1}^{0}{t}^{2} ⋅ {t}^{11}\sqrt{\left(1 + {t}^{5}\right)}\left(t\right)dt$

$⇒ I = 4\mathop{∫}_{0}^{1}{t}^{14}\sqrt{\left(1 + {t}^{5}\right)}dt$

Now, let $1 + {t}^{5} = {k}^{2}$

$⇒ 5{t}^{4}dt = 2kdk$

$∴ I = 4 ⋅ \mathop{∫}_{1}^{\sqrt{2}}{\left({k}^{2} - 1\right)}^{2} ⋅ k \cdot \dfrac{2k}{5}dk$

$⇒ I = \dfrac{8}{5}\mathop{∫}_{1}^{\sqrt{2}}{k}^{6} - 2{k}^{4} + {k}^{2}dk$

$⇒ I = \dfrac{8}{5}{\left[\dfrac{{k}^{7}}{7} - \dfrac{2{k}^{5}}{5} + \dfrac{{k}^{3}}{3}\right]}_{1}^{\sqrt{2}}$

$⇒ I = \dfrac{8}{5}\left[\dfrac{8\sqrt{2}}{7} - \dfrac{8\sqrt{2}}{5} + \dfrac{2\sqrt{2}}{3} - \dfrac{1}{7} + \dfrac{2}{5} - \dfrac{1}{3}\right]$

$⇒ I = \dfrac{8}{5}\left[\dfrac{22\sqrt{2}}{105} - \dfrac{8}{105}\right]$

$⇒ 525 ⋅ I = 176\sqrt{2} - 64$

Hence, on comparing we get, $n = 176$