_ ^ x^3 - x - 2 (1 - x^2 ) ;dx = - Vidyadip

MCQ

$\int_{}^{} {\frac{{{x^3} - x - 2}}{{(1 - {x^2})}}\;dx = } $

A
$\log \left( {\frac{{x + 1}}{{x - 1}}} \right) - \frac{{{x^2}}}{2} + c$
B
$\log \left( {\frac{{x - 1}}{{x + 1}}} \right) + \frac{{{x^2}}}{2} + c$
C
$\log \left( {\frac{{x + 1}}{{x - 1}}} \right) + \frac{{{x^2}}}{2} + c$
✓
$\log \left( {\frac{{x - 1}}{{x + 1}}} \right) - \frac{{{x^2}}}{2} + c$

✓

Answer

Correct option: D.

$\log \left( {\frac{{x - 1}}{{x + 1}}} \right) - \frac{{{x^2}}}{2} + c$

d
(d)$\int_{}^{} {\frac{{{x^3} - x - 2}}{{(1 - {x^2})}}\,dx} = \int_{}^{} {\frac{{ - x(1 - {x^2})}}{{(1 - {x^2})}}\,dx - \int_{}^{} {\frac{2}{{1 - {x^2}}}\,dx} } $
$ = - \int_{}^{} {x\,dx} - 2\int_{}^{} {\frac{1}{{1 - {x^2}}}\,dx = \frac{{ - {x^2}}}{2} + \log \left( {\frac{{x - 1}}{{x + 1}}} \right) + c.} $

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