Questions: ∫ e^x / (e^x + 4) dx

Solution Steps

We start with the integral

\[ \int \frac{e^{x}}{e^{x}+4} \, dx. \]

We use the substitution \(u = e^{x} + 4\), which gives us \(du = e^{x} \, dx\). This transforms our integral into a simpler form.

After substituting, we find that the integral can be expressed as

\[ \int \frac{du}{u} = \ln|u| + C. \]

Substituting back \(u = e^{x} + 4\), we have

\[ \ln|e^{x} + 4| + C. \]

The original integral can also be expressed in terms of \(e^{x}\):

\[ \int \frac{e^{x}}{e^{x}+4} \, dx = e^{x} - 5 \frac{e^{x}}{e^{x} + 4} + C. \]

This leads us to the final expression for the integral.

Final Answer