(x + 3) e^x (x + 4) ^2 , ,dx = , , - Vidyadip

MCQ

$\int {\frac{{(x + 3){e^x}}}{{{{(x + 4)}^2}}}\,\,dx = \,\,} $

A
$\frac{1}{{{{(x + 4)}^2}}} + c$
B
$\frac{{{e^x}}}{{{{(x + 4)}^2}}} + c$
✓
$\frac{{{e^x}}}{{x + 4}} + c$
D
$\frac{{{e^x}}}{{x + 3}} + c$

✓

Answer

Correct option: C.

$\frac{{{e^x}}}{{x + 4}} + c$

c
(c) $I = \int {\frac{{(x + 3){e^x}}}{{{{(x + 4)}^2}}}dx} $$I = \int {\frac{{1 + {{\tan }^2}x}}{{1 - {{\tan }^2}x}}dx} $
$ \Rightarrow I = \int {{e^x}\,\left( {\frac{1}{{x + 4}} - \frac{1}{{{{(x + 4)}^2}}}} \right)\,dx} $
$\therefore I = \frac{{{e^x}}}{{x + 4}} + c$.

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