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Home  >>  JEEMAIN and AIPMT  >>  Mathematics  >>  Class12  >>  Integral Calculus
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Integrate : $ \int \large\frac{x^{e-1}+e^{x-1}}{x^e+e^x}$$dx$

$(a)\;\frac{1}{e} \log _e |x^e +(1/e)^x |+c \qquad(b)\;e \log _e |x^e +e^x |+c\qquad(c)\;\frac{1}{e} \log |x^e +e^x |+c\qquad (d)\;None$
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1 Answer

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Suppose :- $x^e +e^x =t$
differentiate with x
$=> e.x^{e-1}+e^x dx=dt$
$=> e \{ x^{e-1} +e^{x-1}\} dx=dt$
$=> (x^{e-1} +e^{x-1}) dx= \large\frac{dt}{e}$
$=> (x^{e-1}+e^{x-1}) dx =\large\frac{dt}{e}$
By putting values :
$\int \large\frac{1}{e} \times \frac{1}{t} $$dt$
$=>\large\frac{1}{e}$$ \log _e |(t)| +c$
Put the value of t
$=>\large\frac{1}{e}$$ \log |x^e +e^x |+c$
Hence c is the correct answer.
answered Dec 23, 2013 by meena.p
 
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