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
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