Evaluate the following integrals: ^ 2 _0 ( 3+5 3+5 )dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_0\log\Big(\frac{3+5\cos\text{x}}{3+5\sin\text{x}}\Big)\text{dx}$

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Answer

Let, $\text{I}=\int\limits^{\frac{\pi}{2}}_0\log\Big(\frac{3+5\cos\text{x}}{3+5\sin\text{x}}\Big)\text{dx}\ ...(\text{i})$
$=\int\limits^{\frac{\pi}{2}}_0\log\Bigg[\frac{3+5\cos\big(\frac{\pi}{2}-\text{x}\big)}{3+5\sin\big(\frac{\pi}{2}-\text{x}\big)}\Bigg]$
$=\int\limits^{\frac{\pi}{2}}_0\log\Big(\frac{3+5\sin\text{x}}{3+5\cos\text{x}}\Big)\text{dx}\ ...(\text{ii})$
Adding (i) and (ii)
$2\text{I}=\int\limits^{\frac{\pi}{2}}_0\bigg[\log\Big(\frac{3+5\cos\text{x}}{3+5\sin\text{x}}\Big)+\log\Big(\frac{3+5\sin\text{x}}{3+5\cos\text{x}}\Big)\bigg]\text{dx}$
$=\int\limits^{\frac{\pi}{2}}_0\log\Big(\frac{3+5\cos\text{x}}{3+5\sin\text{x}}\times\frac{3+5\sin\text{x}}{3+5\cos\text{x}}\Big)\text{dx}$
$=\int\limits^{\frac{\pi}{2}}_0\log1\text{ dx}=0$
Hence, $\text{I}=0$

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