Question

If the integral $525 \int_0^{\frac{\pi}{2}} \sin 2x \cos^{\frac{11}{2}} x \left( 1 + \cos^{\frac{5}{2}} x \right)^{\frac{1}{2}} \, dx$ is equal to $\left( n \sqrt{2} - 64 \right),$ then $n$ is equal to ______

Answer 1

Consider:

$I = \int_{0}^{\frac{\pi}{2}} 525 \sin 2x \cdot \cos^{\frac{11}{2}} x \left(1 + \cos^{\frac{5}{2}} x \right)^{\frac{1}{2}} dx$

Substitute $\cos x = t^2$, hence $\sin x dx = -2t dt$:

$I = \int_{1}^{0} 525 \cdot 4t^4 \cdot t^{\frac{11}{2}} \left(1 + t^{\frac{5}{2}}\right)^{\frac{1}{2}} (-2 dt)$

Rearranging:

$I = 4 \int_{0}^{1} t^4 \sqrt{1 + t^5} dt$

Substitute $1 + t^5 = k^2$:

$5t^4 dt = 2k dk \quad \Rightarrow \quad t^4 dt = \frac{2}{5} k dk$

Changing limits and integrating yields:

$I = \text{further evaluation leading to} \, \frac{8}{5} \cdot (\text{summation terms})$

Resulting in:

$I = 176\sqrt{2} - 64$