If the integral 525 _0^( )/(2) 2x ^(11)/(2)x(1+Cos^(5)/(2)x)^(1)/(2)dx is equal to (n√2-64…

Question

If the integral $525\int _{0}^{\frac{\pi }{2}}\sin 2x{\cos }^{\frac{11}{2}}x{(1+{Cos}^{\frac{5}{2}}x)}^{\frac{1}{2}}dx$ is equal to $(n\sqrt{2}-64)$, then $n$ is equal to \_\_\_\_\_\_\_\_\_.

Answer: 176

Accepted Answer

$I=\int _{0}^{\frac{\pi }{2}}\sin 2x\cdot (\cos x{)}^{\frac{11}{2}}{(1+(\cos x{)}^{\frac{5}{2}})}^{\frac{1}{2}}dx$

Put $\cos x={t}^{2}\Rightarrow \sin xdx=-2tdt$

$\begin{gathered} 	herefore I=4\int _{0}^{1}{t}^{2}\cdot {t}^{11}\sqrt{(1+{t}^{5})}(t)dt \ I=4\int _{0}^{1}{t}^{14}\sqrt{1+{t}^{5}}dt \end{gathered}$

Put $1+{t}^{5}={k}^{2}$

$\Rightarrow 5{t}^{4}dt=2kdk$

$	herefore I=4\cdot \int _{1}^{\sqrt{2}}{({k}^{2}-1)}^{2}\cdot k\frac{2k}{5}dk$

$I=\frac{8}{5}\int _{1}^{\sqrt{2}}{k}^{6}-2{k}^{4}+{k}^{2}dk$

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

$I=\frac{8}{5}[\frac{8\sqrt{2}}{7}-\frac{8\sqrt{2}}{5}+\frac{2\sqrt{2}}{3}-\frac{1}{7}+\frac{2}{5}-\frac{1}{3}]$

$\begin{gathered} I=\frac{8}{5}[\frac{22\sqrt{2}}{105}-\frac{8}{105}] \ 	herefore 525\cdot I=176\sqrt{2}-64 \end{gathered}$

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