_ ^ dx x( x^7 + 1) = - Vidyadip

MCQ

$\int_{}^{} {\frac{{dx}}{{x({x^7} + 1)}}} = $

A
$\log \left( {\frac{{{x^7}}}{{{x^7} + 1}}} \right) + c$
✓
$\frac{1}{7}\log \left( {\frac{{{x^7}}}{{{x^7} + 1}}} \right) + c$
C
$\log \left( {\frac{{{x^7} + 1}}{{{x^7}}}} \right) + c$
D
$\frac{1}{7}\log \left( {\frac{{{x^7} + 1}}{{{x^7}}}} \right) + c$

✓

Answer

Correct option: B.

$\frac{1}{7}\log \left( {\frac{{{x^7}}}{{{x^7} + 1}}} \right) + c$

b
(b) Given, $\int_{}^{} {\frac{{dx}}{{x\,({x^7} + 1)}}} = \int_{}^{} {\frac{{dx}}{{{x^8}\left( {1 + \frac{1}{{{x^7}}}} \right)}}} $
Put $1 + \frac{1}{{{x^7}}} = t$ ==> $\frac{{ - 7}}{{{x^8}}}dx = dt$
$I = \frac{{ - 1}}{7}\int {\frac{{dt}}{t} = } \frac{{ - 1}}{7}\log t + c$
==> $I = - \frac{1}{7}\log \left( {\frac{{{x^7} + 1}}{{{x^7}}}} \right) + c$ ==> $I = \frac{1}{7}\log \left( {\frac{{{x^7}}}{{{x^7} + 1}}} \right) + c$.

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