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7.1 inderfinite integral — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths7.1 inderfinite integral1 Mark
MCQ
$\sqrt 2 \smallint \frac{{sinx\;dx}}{{{\rm{sin}}\left( {x - \frac{\pi }{4}} \right)}} = $
  • A
    $x + \log \left| {\cos \left( {x - \frac{\pi }{4}} \right)} \right| + c\;$
  • B
    $\;x - \log \left| {\sin \left( {x - \frac{\pi }{4}} \right)} \right| + c$
  • ✓
    $\;x + \log \left| {\sin \left( {x - \frac{\pi }{4}} \right)} \right| + c$
  • D
    $\;x - \log \left| {\cos \left( {x - \frac{\pi }{4}} \right)} \right| + c$
✓

Answer

Correct option: C.
$\;x + \log \left| {\sin \left( {x - \frac{\pi }{4}} \right)} \right| + c$
c
$\sqrt{2} \int \frac{\sin x d x}{\sin \left(x-\frac{\pi}{4}\right)}$$=\sqrt{2} \int \frac{\sin \left(x-\frac{\pi}{4}+\frac{\pi}{4}\right) d x}{\sin \left(x-\frac{\pi}{4}\right)}$

$=\sqrt{2} \int\left(\cos \frac{\pi}{4}+\cot \left(x-\frac{\pi}{4}\right) \sin \frac{\pi}{4}\right) d x$

$=\int d x+\int \cot \left(x-\frac{\pi}{4}\right) d x$

$=x+\ln \left|\sin \left(x-\frac{\pi}{4}\right)\right|+c$

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