Determine the integral by making an appropriate substitution.
-$\int \frac { \mathrm { e } ^ { \mathrm { x } } } { 1 + 2 \mathrm { e } ^ { \mathrm { x } } } \mathrm { dx }$
A) $\mathrm { e } ^ { 2 x }$ + C
B) $\left( 1 + 2 e ^ { x } \right) ^ { 2 }$ + C
C) ln $\left| 1 + 2 \mathrm { e } ^ { x } \right|$ + C
D) $\frac { 1 } { 2 }$ ln $\left| 1 + 2 \mathrm { e } ^ { x } \right|$ + C
E) none of these

Question

Determine the integral by making an appropriate substitution. 
-$\int \frac { \mathrm { e } ^ { \mathrm { x } } } { 1 + 2 \mathrm { e } ^ { \mathrm { x } } } \mathrm { dx }$
A) $\mathrm { e } ^ { 2 x }$ + C
B) $\left( 1 + 2 e ^ { x } \right) ^ { 2 }$ + C
C) ln $\left| 1 + 2 \mathrm { e } ^ { x } \right|$ + C
D) $\frac { 1 } { 2 }$ ln $\left| 1 + 2 \mathrm { e } ^ { x } \right|$ + C
E) none of these

Quizplus · Accepted Answer

Using the substitution $ u = 1 + 2e^x $, the integral simplifies to $\frac{1}{2} \int \frac{1}{u} \, du$, which results in $\frac{1}{2} \ln |u| + C$. Substituting back gives $\frac{1}{2} \ln |1 + 2e^x| + C$.

Determine the Integral by Making an Appropriate Substitution ∫ex1+2exdx\int \frac { \mathrm { e } ^ { \mathrm { x } } } { 1 + 2 \mathrm { e } ^ { \mathrm { x } } } \mathrm { dx }∫1+2exex​dx

Determine the Integral by Making an Appropriate Substitution $\int \frac { \mathrm { e } ^ { \mathrm { x } } } { 1 + 2 \mathrm { e } ^ { \mathrm { x } } } \mathrm { dx }$