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STD 12 - 7.1 inderfinite integral — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 7.1 inderfinite integral1 Mark
MCQ
$\int_{}^{} {\frac{{dx}}{{x({x^5} + 1)}}} = $
  • A
    $\frac{1}{5}\log {x^5}({x^5} + 1) + c$
  • B
    $\frac{1}{5}\log {x^5}\left( {\frac{{1 + {x^5}}}{{{x^5}}}} \right) + c$
  • C
    $\frac{1}{5}\log {x^5}\left( {\frac{{{x^5}}}{{{x^5} + 1}}} \right) + c$
  • ✓
    None of these
✓

Answer

Correct option: D.
None of these
d
(d) We have $I = \int {\frac{{dx}}{{x({x^5} + 1)}}} = \int {\frac{{dx}}{{{x^6}\left( {1 + \frac{1}{{{x^5}}}} \right)}}} $
Put $1 + \frac{1}{{{x^5}}} = t$ ==> $\frac{{ - 5}}{{{x^6}}}dx = dt$
==> $I = - \frac{1}{5}\int {\frac{{dt}}{t} = - \frac{1}{5}} \log t + c$
$I = - \frac{1}{5}\log \left( {1 + \frac{1}{{{x^5}}}} \right) + c = - \frac{1}{5}\log \left( {\frac{{{x^5} + 1}}{{{x^5}}}} \right) + c$
 $I = \frac{1}{5}\log \left( {\frac{{{x^5}}}{{{x^5} + 1}}} \right) + c$.

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