The correct option is A log|e^x+2/e^x+3|+c e^x= t e^xdx= dt ∫dtt^2+5t+6= ∫dt(t+3)(t+2) = ∫ [ 1(t+2) 1(t+3) ]dt = ∫1(t+2)dt ∫1(t+3 ...

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Integration by Partial Fractions

∫ex/e2x+5ex+6...

Question

$\int \frac{e^{x}}{e^{2 x} + 5 e^{x} + 6} d x =$

log | \frac{e^{x} + 2}{e^{x} + 3} | + c

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log | \frac{e^{x} + 3}{e^{x} + 2} | + c

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log | \frac{e^{x} - 2}{e^{x} - 3} | + c

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log | \frac{e^{x} - 3}{e^{x} - 2} | + c

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Solution

The correct option is A $log | \frac{e^{x} + 2}{e^{x} + 3} | + c$
$e^{x} = t$
$e^{x} d x = d t$
$\int \frac{d t}{t^{2} + 5 t + 6} = \int \frac{d t}{(t + 3) (t + 2)}$
$= \int [\frac{1}{(t + 2)} - \frac{1}{(t + 3)}] d t$
$= \int \frac{1}{(t + 2)} d t - \int \frac{1}{(t + 3)} d t$
$= log (t + 2) - log (t + 3) + c$
$= log \frac{(t + 2)}{(t + 3)} + c$
$= log \frac{(e^{x} + 2)}{(e^{x} + 3)} + c$

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