( x + 2 x + 4 ) ^2 e^x , ,dx is equal to - Vidyadip

MCQ

$\int {{{\left( {\frac{{x + 2}}{{x + 4}}} \right)}^2}{e^x}\,\,dx} $ is equal to

✓
${e^x}\left( {\frac{x}{{x + 4}}} \right) + c$
B
${e^x}\left( {\frac{{x + 2}}{{x + 4}}} \right) + c$
C
${e^x}\left( {\frac{{x - 2}}{{x + 4}}} \right) + c$
D
$\left( {\frac{{2x{e^x}}}{{x + 4}}} \right) + c$

✓

Answer

Correct option: A.

${e^x}\left( {\frac{x}{{x + 4}}} \right) + c$

a
(a) $I = \int {{{\left( {\frac{{x + 2}}{{x + 4}}} \right)}^2}{e^x}dx} $$ = \int {{e^x}\left[ {\frac{{{x^2} + 4x + 4}}{{{{(x + 4)}^2}}}} \right]} \,dx$
$ \Rightarrow I = \int {{e^x}\left[ {\frac{{x(x + 4)}}{{{{(x + 4)}^2}}} + \frac{4}{{{{(x + 4)}^2}}}} \right]\,dx} $
$ = {e^x}\left[ {\frac{x}{{x + 4}} + \frac{4}{{{{(x + 4)}^2}}}} \right]\,dx$$ = {e^x}\left( {\frac{x}{{x + 4}}} \right) + c$.

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