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Q.
Let $a_n= \int\limits_{0} ^{\pi/2}\cos^n\,x\,\cos\,nx\,dx$ . Thus $a_n:a_{n+1}$ is equal to
Integrals
Solution:
$a_{n}=\int_{0}^{\frac{\pi}{2}} \cos ^{n} x \cos n x d x$
$=\left[\cos ^{n} x \cdot \frac{\sin n x}{n}\right]_{0}^{\frac{\pi}{2}}-\int_{0}^{\frac{\pi}{2}} n \cos ^{n-1} x(-\sin x) \cdot \frac{\sin n x}{n} d x$
$=0+\int_{0}^{\frac{\pi}{2}} \cos ^{n-1} x \sin x \sin n x d x$
$=\int_{0}^{\frac{\pi}{2}} \cos ^{n-1} x \cos (n-1) x-\int_{0}^{\frac{\pi}{2}} \cos ^{n} x \cos n x \quad d x$
$=a_{n-1}-a_{n}$
$a_{n}=\frac{1}{2} \Rightarrow \frac{a_{n}+1}{a_{n}}=\frac{1}{2}$