MCQ
$\int\limits_{\frac{\pi }{4}}^{\frac{\pi }{4}} {{{\log }_e}\left( {\sin x + \cos x} \right)\,dx}$ is equal to
- ✓$ - \frac{\pi }{4}\ln 2$
- B$ \frac{\pi }{4}\ln 2$
- C$ \frac{\pi }{8}\ln 2$
- D$ -\frac{\pi }{8}\ln 2$
Putting $\left(\mathrm{x}+\frac{\pi}{4}\right)=\theta ; \mathrm{dx}=\mathrm{d} \theta$
$=\int_{0}^{\frac{\pi}{2}} \log (\sqrt{2} \sin \theta) d \theta$
$=\int_{0}^{\frac{\pi}{2}} \log \sqrt{2} \mathrm{d} \theta+\int_{0}^{\frac{\pi}{2}} \log \sin \theta \mathrm{d} \theta$
$=\log \sqrt{2}(\theta)_{0}^{\pi / 2}-\frac{\pi}{2} \ln 2$
$=\frac{\pi}{4} \ln 2-\frac{\pi}{2} \ln 2=-\frac{\pi}{4} \ln 2$
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