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Question

Find: $\int\sin\text{x}.\log\cos\text{x}\text{dx}.$

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Answer

$\int\sin\text{x}.\log\cos\text{x}\text{dx}=$
$\log\cos\text{x}\int\sin\text{x}\text{dx}-\int\Big(\frac{\text{d}}{\text{dx}}(\log\cos\text{x})\int\sin\text{x}\text{dx}\Big)\text{dx}$
$=-\cos\text{x}\log\cos\text{x}-\int\Big(\frac{\sin\text{x}}{\cos\text{x}}\times\cos\text{x}\Big)\text{dx}=-\cos\text{x}\log\cos\text{x}+\cos\text{x}+\text{C}.$

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