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

MCQ

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

A
$ - \sqrt 2 {\tan ^{ - 1}}x + \sqrt 3 {\tan ^{ - 1}}x + c$
✓
$ - \sqrt 2 {\tan ^{ - 1}}\frac{x}{{\sqrt 2 }} + \sqrt 3 {\tan ^{ - 1}}\frac{x}{{\sqrt 3 }} + c$
C
$\sqrt 2 {\tan ^{ - 1}}\frac{x}{{\sqrt 2 }} + \sqrt 3 {\tan ^{ - 1}}\frac{x}{{\sqrt 3 }} + c$
D
None of these

✓

Answer

Correct option: B.

$ - \sqrt 2 {\tan ^{ - 1}}\frac{x}{{\sqrt 2 }} + \sqrt 3 {\tan ^{ - 1}}\frac{x}{{\sqrt 3 }} + c$

b
(b)$\int_{}^{} {\frac{{{x^2}}}{{({x^2} + 2)({x^2} + 3)}}} \,dx = \int_{}^{} {\left[ {\frac{3}{{{x^2} + 3}} - \frac{2}{{{x^2} + 2}}} \right]} \,dx$
$ = \frac{3}{{\sqrt 3 }}{\tan ^{ - 1}}\frac{x}{{\sqrt 3 }} - \frac{2}{{\sqrt 2 }}{\tan ^{ - 1}}\left( {\frac{x}{{\sqrt 2 }}} \right) + c$
$ = \sqrt 3 {\tan ^{ - 1}}\left( {\frac{x}{{\sqrt 3 }}} \right) - \sqrt 2 {\tan ^{ - 1}}\left( {\frac{x}{{\sqrt 2 }}} \right) + c.$

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