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7.1 inderfinite integral — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths7.1 inderfinite integral1 Mark
MCQ
$\int_{}^{} {\frac{1}{{\sqrt {1 + \sin x} }}dx} = $
  • A
    $2\sqrt 2 \log \tan \left( {\frac{\pi }{8} + \frac{x}{4}} \right) + c$
  • B
    $\frac{1}{{\sqrt 2 }}\log \tan \left( {\frac{\pi }{8} + \frac{x}{4}} \right) + c$
  • ✓
    $\sqrt 2 \log \tan \left( {\frac{\pi }{8} + \frac{x}{4}} \right) + c$
  • D
    $\frac{1}{{2\sqrt 2 }}\log \tan \left( {\frac{\pi }{8} + \frac{x}{4}} \right) + c$
✓

Answer

Correct option: C.
$\sqrt 2 \log \tan \left( {\frac{\pi }{8} + \frac{x}{4}} \right) + c$
c
(c)$\int_{}^{} {\frac{1}{{\sqrt {1 + \sin x} }}} \,dx = \int_{}^{} {\frac{1}{{\sqrt 2 \sin \left( {\frac{\pi }{4} + \frac{x}{2}} \right)}}} \,dx$
$ = \frac{1}{{\sqrt 2 }}\int_{}^{} {{\rm{cosec}}\,\left( {\frac{x}{{\rm{2}}} + \frac{\pi }{4}} \right)} \,dx = \sqrt 2 \log \tan \left( {\frac{\pi }{8} + \frac{x}{4}} \right) + c.$

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