(1 + x - x^ - 1 ) e^ x + x^ - 1 , ,dx = - Vidyadip

MCQ

$\int {(1 + x - {x^{ - 1}}){e^{x + {x^{ - 1}}}}\,\,dx = } $

A
$(x + 1){e^{x + {x^{ - 1}}}} + c$
B
$(x - 1){e^{x + {x^{ - 1}}}} + c$
C
$ - x{e^{x + {x^{ - 1}}}} + c$
✓
$x{e^{x + {x^{ - 1}}}} + c$

✓

Answer

Correct option: D.

$x{e^{x + {x^{ - 1}}}} + c$

d
(d) $\int {(1 + x - {x^{ - 1}}){e^{x + {x^{ - 1}}}}dx} $
$ = \int {x\,{e^{x + {x^{ - 1}}}}\left( {1 - \frac{1}{{{x^2}}}} \right) + {e^{x + {x^{ - 1}}}}]\,dx} $
$\left(\because {\int {[x\,f'(x) + f(x)]dx = x\,f(x) + c} } \right)$
$\therefore \int {(1 + x - {x^{ - 1}}){e^{x + {x^{ - 1}}}}dx = x{e^{x + {x^{ - 1}}}}} + c$.

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