A) \[\frac{1}{{{(x+4)}^{2}}}+c\]
B) \[\frac{{{e}^{x}}}{{{(x+4)}^{2}}}+c\]
C) \[\frac{{{e}^{x}}}{x+4}+c\]
D) \[\frac{{{e}^{x}}}{x+3}+c\]
Correct Answer: C
Solution :
\[\int_{{}}^{{}}{{{e}^{x}}\left[ \frac{(x+4)-1}{{{(x+4)}^{2}}} \right]}\,dx\] \[=\int_{{}}^{{}}{{{e}^{x}}.\frac{1}{(x+4)}dx-\int_{{}}^{{}}{\frac{{{e}^{x}}}{{{(x+4)}^{2}}}}dx}\] \[f(x)=\frac{1}{x+4}\]and \[f'(x)=-\frac{1}{{{(x+4)}^{2}}}\] \[I=\int_{{}}^{{}}{{{e}^{x}}[f(x)+f'(x)]}dx={{e}^{x}}f(x)+c\] \[=\frac{{{e}^{x}}}{x+4}+c\]
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