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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$f(x)=\left[\begin{array}{ll}{\left[e^{x}\right],} \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,x<0 \\ a e^{x}+[x-1], \,\,\,\,\,\,\,\,\,0 \leq x<1 \\ b+[\sin (\pi x)], \,\,\,\,\,\,\,\,\,\,\,\,1 \leq x<2 \\ {\left[e^{-x}\right]-c,} \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,x \geq 2\end{array}\right.$
where a,b,c $\in R$ and $[t]$ denotes greatest integer less than or equal to $t.$ Then, which of the following statements is true $?$