Doubt from integration

ques 11,12
@Ajay_2020_1 @Sneha_2021 @akash_2020 anyone

Easiest way is to solve I1,I2,I3

Yeah bro that's my first approach to the problem . But i wanted to know the exact solution

@Hari_Shankar sir

Method 1 is \color{red} \textbf{ residue theoram}
Method 2
Find I_{n-1}
Open formula of cos(n-1)x
Now after that put y=sin(nx)
U will get result
@Achyut_2020

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  • In lst step just compared 1st term because other part will be imaginary ans is real part
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I_n=\displaystyle \int_0^{\frac{\pi}{2}} \cos^n x \cos nx \ \mathrm{dx}= \int_0^{\frac{\pi}{2}} \cos^n x [\cos (n+1)x \cos x + \sin(n+1)x \sin x] \ \mathrm{dx}

=\underbrace{\displaystyle \int_0^{\frac{\pi}{2}} \cos^{n+1} x \cos (n+1)x\ \mathrm{dx}}_{I_{n+1}}+\displaystyle \int_0^{\frac{\pi}{2}} \cos^n x \sin(n+1)x \sin x \ \mathrm{dx}

Now, \displaystyle \int_0^{\frac{\pi}{2}} \cos^n x \sin(n+1)x \sin x \ \mathrm{dx}= -\left|\dfrac{1}{n+1}\cos^{n+1}x \sin (n+1)x\right|_0^{\frac{\pi}{2}}+ \int_0^{\frac{\pi}{2}} \cos^{n+1} x \cos (n+1)x\ \mathrm{dx}

=\displaystyle \int_0^{\frac{\pi}{2}} \cos^{n+1} x \cos (n+1)x\ \mathrm{dx}=I_{n+1}

Thus we get I_n=I_{n+1}+I_{n+1}=2I_{n+1}

Thus the series \{I_k\} is a GP with common ration \dfrac{1}{2}

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