Question

If $\underset{0}{\overset{\pi }{\int }}\frac{5^{\cos x}\left(1+\cos x\cos 3x+{\cos }^{2}x+{\cos }^{3}x\cos 3x\right)dx}{1+5^{\cos x}}=\frac{k\pi }{16}$, then $k$ is equal to

Answer 1

Answer: 26

$I=\underset{0}{\overset{\pi }{\int }}\frac{5^{\cos x}\left(1+\cos x\cos 3x+{\cos }^{2}x+{\cos }^{3}x\cos 3x\right)}{1+5^{\cos x}}dx$
$I=\underset{0}{\overset{\pi }{\int }}\frac{5^{-\cos x}\left(1+\cos x\cos 3x+{\cos }^{2}x+{\cos }^{3}x\cos 3x\right)}{1+5^{-\cos x}}dx$
$2I=\underset{0}{\overset{\pi }{\int }}\left(1+\cos x\cos 3x+{\cos }^{2}x+{\cos }^{3}x\cos 3x\right)dx$
$/2I=/2\underset{0}{\overset{\frac{\pi }{2}}{\int }}\left(1+\cos x\cos 3x+{\cos }^{2}x+{\cos }^{3}x\cos 3x\right)dx$
$I=\underset{0}{\overset{\frac{\pi }{2}}{\int }}\left(1+\sin x(-\sin 3x)+{\sin }^{2}x-{\sin }^{3}x\sin 3x\right)dx$
$2I=\underset{0}{\overset{\frac{\pi }{2}}{\int }}\left(3+\cos 4x+{\cos }^{3}x\cos 3x-{\sin }^{3}x\sin 3x\right)dx$
$2I=\underset{0}{\overset{\frac{\pi }{2}}{\int }}3+\cos 4x+\left(\frac{\cos 3x+3\cos x}{4}\right)\cos 3x-\sin 3x\left(\frac{3\sin x-\sin 3x}{4}\right)dx$
$2I=\underset{0}{\overset{\frac{\pi }{2}}{\int }}\left(3+\cos 4x+\frac{1}{4}+\frac{3}{4}\cos 4x\right)dx$
$2I=\frac{13}{4}\times \frac{\pi }{2}+\frac{7}{4}{\left(\frac{\sin 4x}{4}\right)}_0^{\frac{\pi }{2}}\Rightarrow I=\frac{13\pi }{16}$