_ ^ dx x( x^5 + 1) = - Vidyadip

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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