∫ ((x^4 e^x dx)/((x^4 +4x^3 +12x^2 +24x+24+72e^x )^2 )) = ?


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Question Number 144359 by mathdanisur last updated on 24/Jun/21

$\int \frac{x^{4} e^{x} d x}{{(x^{4} + 4 x^{3} + 12 x^{2} + 24 x + 24 + 72 e^{x})}^{2}} = ?$
	Answered by mitica last updated on 24/Jun/21

	$f (x) = x^{4} + 4 x^{3} + 12 x^{2} + 24 x + 24 + 72 e^{x}$ $f^{'} (x) = 4 x^{3} + 12 x^{2} + 24 x + 24 + 72 e^{x}$ $f (x) - f^{'} (x) = x^{4}$ $\int \frac{f (x) - f^{'} (x)}{f^{2} (x)} e^{x} d x = \int \frac{e^{x}}{f (x)} d x + \int {(\frac{1}{f (x)})}^{'} e^{x} =$ $\int \frac{e^{x}}{f (x)} + e^{x} \cdot \frac{1}{f (x)} - \int \frac{e^{x}}{f (x)} d x = \frac{e^{x}}{f (x)} + c$
	Commented by mathdanisur last updated on 24/Jun/21

	$T h a n k s S i r, a n s w e r : x^{4} . ?$
	Commented by mathdanisur last updated on 24/Jun/21

	$a l o t p e r f e c t s o l u t i o n t h a n k y o u S i r$
	Commented by mitica last updated on 24/Jun/21

	$\frac{e^{x}}{f (x)}$
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