Question Bank [2022] — Maths STD 12 Science — Question
Maharashtra BoardEnglish MediumSTD 12 ScienceMathsQuestion Bank [2022]2 Marks
Question
$\int[\operatorname{cosec}(\log x)][1-\cot (\log x)] d x$
✓
Answer
$ \text { Let } I =\int[\operatorname{cosec}(\log x)][1-\cot (\log x)] d x$
$\text { Put } \log _{ e } x = t$
$\therefore x = e ^{ t }$
$\therefore dx = e ^{ t } \cdot dt$
$\therefore I =\int \operatorname{cosec} t (1-\cot t ) e ^{ t } dt$
$=\int e ^{ t }(\operatorname{cosec} t -\operatorname{cosec} t \cdot \cot t ) dt $
Put $f(t)=\operatorname{cosec} t$
$ \therefore f ^{\prime}( t )=-\operatorname{cosec} t \cdot \cot t$
$\therefore I =\int e ^{ t }\left[ f ( t )+ f ^{\prime}( t )\right] dt$
$= e ^{ t } \cdot f ( t )+ c = e ^{ t } \operatorname{cosec} t + c$
$\therefore I =x \operatorname{cosec}(\log x)+ c $
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