_ ^ cosec - cosec + ;d = - Vidyadip

MCQ

$\int_{}^{} {\frac{{{\rm{cosec}}\theta - \cot \theta }}{{{\rm{cosec}}\theta + \cot \theta }}} \;d\theta = $

✓
$2{\rm{cosec}}\theta - 2\cot \theta - \theta + c$
B
$2\,{\rm{cosec}}\theta - 2\cot \theta + \theta + c$
C
$2\,{\rm{cosec}}\theta + 2\cot \theta - \theta + c$
D
None of these

✓

Answer

Correct option: A.

$2{\rm{cosec}}\theta - 2\cot \theta - \theta + c$

a
(a)$\int_{}^{} {\frac{{{\rm{cosec}}\theta - \cot \theta }}{{{\rm{cosec}}\theta + \cot \theta }}\,d\theta } = \int_{}^{} {{{({\rm{cosec}}\theta - \cot \theta )}^2}d\theta } $
$ = \int_{}^{} {{\rm{cose}}{{\rm{c}}^2}\theta \,d\theta } + \int_{}^{} {{{\cot }^2}\theta \,d\theta } - 2\int_{}^{} {{\rm{cosec}}\theta \cot \theta \,d\theta } $
$ = \int_{}^{} {(2{\rm{cose}}{{\rm{c}}^2}\theta - 1)\,d\theta } - 2\int_{}^{} {{\rm{cosec}}\theta \cot \theta \,d\theta } $
$ = 2{\rm{cosec}}\theta - 2\cot \theta - \theta + c.$

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