Use a trigonometric substitution to evaluate the integral.
-$$\int \frac { e ^ { x } d x } { \sqrt { 1 - e ^ { 2 x } } }$$
A) $\sin ^ { - 1 } \left( e ^ { x } \right) + C$
B) $- 2 \sqrt { 1 - \mathrm { e } ^ { 2 \mathrm { x } } } + \mathrm { C }$
C) $e ^ { x } \sin ^ { - 1 } \left( e ^ { x } \right) + C$
D) $\sec ^ { - 1 } \left( \mathrm { e } ^ { \mathrm { x } } \right) + \mathrm { C }$

Question

Quizplus · Accepted Answer

The integral $\int \frac { e ^ { x } d x } { \sqrt { 1 - e ^ { 2 x } } }$ can be solved using a trigonometric substitution. Let $e^x = \sin(\theta)$, then $e^{2x} = \sin^2(\theta)$, and $dx = \frac{d\theta}{\cos(\theta)}$. Substituting these into the integral simplifies it to $\int d\theta = \theta + C$, which is $\sin^{-1}(e^x) + C$.

Use a Trigonometric Substitution to Evaluate the Integral ∫exdx1−e2x\int \frac { e ^ { x } d x } { \sqrt { 1 - e ^ { 2 x } } }∫1−e2x​exdx​

Use a Trigonometric Substitution to Evaluate the Integral $\int \frac { e ^ { x } d x } { \sqrt { 1 - e ^ { 2 x } } }$