MCQ
$\int_{\pi {\rm{/4}}}^{\pi {\rm{/2}}} {{e^x}(\log \sin x + \cot x)\,dx = } $
- A${e^{\pi /4}}\log 2$
- B$ - {e^{\pi /4}}\log 2$
- ✓$\frac{1}{2}{e^{\pi /4}}\log 2$
- D$ - \frac{1}{2}{e^{\pi /4}}\log 2$
$I = \int_{\pi /4}^{\pi /2} {{e^x}\log \sin x\,dx + \int_{\pi /4}^{\pi /2} {{e^x}\cot x\,dx} } $
$ = \int_{\pi /4}^{\pi /2} {{e^x}\log \sin xdx + [{e^x}\log \sin x]_{\pi /4}^{\pi /2}} $$ - \int_{\pi /4}^{\pi /2} {{e^x}\log \sin x\,dx} $
$ = {e^{\pi /2}}\log \sin \frac{\pi }{2} - {e^{\pi /4}}\log \sin \frac{\pi }{4} $
$= \frac{1}{2}{e^{\pi /4}}\log 2$.
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Statement $-2$ : $f(x) = \frac{1}{{\sqrt {1 - {x^2}} }} + \left[ {\frac{{{x^2} + x + 1}}{4}} \right]$ , where $[.]$ is greatest integer function. Function $f(x)$ is even function