Mathematics

Solve:$\int\frac{{{e^x}}}{{\left( {1 + {e^x}} \right)\left( {2 + {e^x}} \right)}}$ dx

SOLUTION
$I=\displaystyle \int \dfrac {e^x dx}{(1+e^x)(2+e^x)}$
put $e^x=t$
$e^x\ dx=dt$
$=\displaystyle \int \dfrac {dt}{(1+t)(2+t)}=\displaystyle \int dt \left [\dfrac {(2+t)-(1+t)}{(1+t)(2+t)}\right]$
$=\displaystyle \int dt \left [\dfrac {1}{(1+t)} - \dfrac {1}{(2+t)}\right]$
$I=\displaystyle \int \dfrac {dt}{t+1}-\displaystyle \int \dfrac {dt}{(2+t)}$
$I=\ln |t+1|-\ln |t+2|+\ln |c|$
$I=\ln \left |c \left (\dfrac {t+1}{t+2}\right )\right|\quad c=$ constant
$I=\ln \left |c \left (\dfrac {e^x+1}{e^x+2}\right )\right|$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

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