Evaluate the following integrals: ^ _2 (1- )dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^{\pi}_2\log(1-\cos\text{x})\text{dx}$

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Answer

Let $\text{I}=\int\limits^{\pi}_2\log(1-\cos\text{x})\text{dx}$
$=\int\limits^{\pi}_2\Big(2\sin^2\frac{\text{x}}{2}\Big)\text{dx}$
$=\int\limits^{\pi}_2\log2\text{ dx}+\int\limits^{\pi}_2\log\sin\frac{\text{x}}{2}\text{ dx}$
Let $\text{t}=\frac{\text{x}}{2}$ in these cong integral then $\text{dt}=\frac{1}{2}\text{ dx}$
When $\text{x}\rightarrow0;\text{t}\rightarrow0$ and $\text{x}\rightarrow\pi;\text{t}\rightarrow\frac{\pi}{2}$
$\text{I}=\log2\big[\text{x}\big]^{\pi}_0+4\int\limits^{\frac{\pi}{2}}_0\log\sin\text{t dt}$
$=\pi\log2+4\times\Big(-\frac{\pi}{2}\log2\Big)$ $\Bigg[\text{Where,}\int\limits^{\frac{\pi}{2}}_0\log\sin\text{t dt}=-\frac{\pi}{2}\log2\Bigg]$
$=-\pi\log2$

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