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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{{\sin 2xdx}}{{1 + {{\cos }^2}x}}} = $
  • A
    $\frac{1}{2}\log (1 + {\cos ^2}x) + c$
  • B
    $2\log (1 + {\cos ^2}x) + c$
  • C
    $\frac{1}{2}\log (1 + \cos 2x) + c$
  • ✓
    $ - \log (1 + {\cos ^2}x) + c$
✓

Answer

Correct option: D.
$ - \log (1 + {\cos ^2}x) + c$
d
(d) $I = \int {\frac{{\sin 2x}}{{1 + {{\cos }^2}x}}dx = \int {\frac{{2\sin x\cos x}}{{1 + {{\cos }^2}x}}dx} } $
Put $1 + {\cos ^2}x = t$ ==> $ - 2\sin x\cos x\,\,dx = dt$
==> $\sin 2x = - dt$. Hence

$I = \int {^ - \left( {\frac{{dt}}{t}} \right)} = - \log t + c$
$ = - \log (1 + {\cos ^2}x) + c$.

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