_ ^ dx 1 + x + x^2 + x^3 = - Vidyadip

MCQ

$\int_{}^{} {\frac{{dx}}{{1 + x + {x^2} + {x^3}}} = } $

✓
$\log \sqrt {1 + x} - \frac{1}{2}\log \sqrt {1 + {x^2}} + \frac{1}{2}{\tan ^{ - 1}}x + c$
B
$\log \sqrt {1 + x} - \log \sqrt {1 + {x^2}} + {\tan ^{ - 1}}x + c$
C
$\log \sqrt {1 + {x^2}} - \log \sqrt {1 + x} + \frac{1}{2}{\tan ^{ - 1}}x + c$
D
$\log \sqrt {1 + x} + {\tan ^{ - 1}}x + \log \sqrt {1 + {x^2}} + c$

✓

Answer

Correct option: A.

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

a
(a)$\int_{}^{} {\frac{{dx}}{{1 + x + {x^2} + {x^3}}} = \int_{}^{} {\frac{{dx}}{{(1 + x)(1 + {x^2})}}} } $
$ = \frac{1}{2}\int_{}^{} {\frac{1}{{1 + {x^2}}}\,dx} + \frac{1}{2}\int_{}^{} {\frac{1}{{1 + x}}\,dx} - \frac{1}{2}\int_{}^{} {\frac{x}{{1 + {x^2}}}\,dx} $
$ = \frac{1}{2}{\tan ^{ - 1}}x + \log \sqrt {1 + x} - \frac{1}{2}\log \sqrt {1 + {x^2}} + c$.

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