Evaluate the integral by making a substitution and then using a table of integrals.
-$$\int e ^ { x } \sqrt { 25 - e ^ { 2 x } } d x$$
A) $\frac { \mathrm { e } ^ { \mathrm { x } } } { 2 } \sqrt { 25 - \mathrm { e } ^ { 2 \mathrm { x } } } + \frac { 25 } { 2 } \sin ^ { - 1 } \left( \frac { \mathrm { e } ^ { \mathrm { x } } } { 5 } \right) + \mathrm { C }$
B) $\frac { e ^ { x } } { 2 } \sqrt { 25 - e ^ { 2 x } } + \frac { 25 } { 2 } \ln \left| e ^ { x } + \sqrt { e ^ { 2 x } - 25 } \right| + C$
C) $\frac { 25 } { 2 } \sin ^ { - 1 } \left( \frac { e ^ { x } } { 5 } \right) + C$
D) $\frac { \mathrm { e } ^ { \mathrm { x } } } { 2 } \sqrt { 25 - \mathrm { e } ^ { 2 \mathrm { x } } } + \frac { 25 } { 2 } \ln \left| \mathrm { x } + \sqrt { \mathrm { x } ^ { 2 } - 25 } \right| + \mathrm { C }$

Question

Quizplus · Accepted Answer

The substitution $u = e^x$ simplifies the integral to a form involving $\sqrt{25-u^2}$, which leads to an answer involving $\sin^{-1}\left(\frac{u}{5}\right)$ and the original function under the square root, matching option A.

Evaluate the Integral by Making a Substitution and Then Using ∫ex25−e2xdx\int e ^ { x } \sqrt { 25 - e ^ { 2 x } } d x∫ex25−e2x​dx

Evaluate the Integral by Making a Substitution and Then Using $\int e ^ { x } \sqrt { 25 - e ^ { 2 x } } d x$