Questions: Find the indefinite integral. (Remember the constant of integration.) ∫ e^x √(1-e^(2x)) dx

Solution Steps

To solve the indefinite integral \(\int e^{x} \sqrt{1-e^{2 x}} \, dx\), we can use a substitution method. Let \( u = 1 - e^{2x} \), then \( du = -2e^{2x} \, dx \). This substitution will simplify the integral into a form that is easier to integrate.

We start with the integral

\[ \int e^{x} \sqrt{1-e^{2 x}} \, dx. \]

Using the substitution \( u = 1 - e^{2x} \), we find that

\[ du = -2e^{2x} \, dx \quad \Rightarrow \quad dx = -\frac{du}{2e^{2x}}. \]

This substitution simplifies the integral.

Rewriting \( e^{2x} \) in terms of \( u \):

\[ e^{2x} = 1 - u \quad \Rightarrow \quad e^{x} = \sqrt{1 - u}. \]

Substituting these into the integral gives:

\[ \int e^{x} \sqrt{1-e^{2 x}} \, dx = \int \sqrt{1 - u} \sqrt{u} \left(-\frac{du}{2(1-u)}\right). \]

This integral can be simplified further, but we will directly evaluate the original integral.

The result of the integral is:

\[ \int e^{x} \sqrt{1-e^{2 x}} \, dx = \frac{1}{2} \sqrt{1 - e^{2x}} e^{x} + \frac{1}{2} \arcsin(e^{x}) + C, \]

where \( C \) is the constant of integration.

Final Answer