Sure, let's solve the integral [tex]\(\int e^{-x}\left(1+e^{2 x}\right) \, dx\)[/tex].
### Step-by-Step Solution:
1. Expand the Integrand:
First, distribute [tex]\(e^{-x}\)[/tex] inside the parentheses:
[tex]\[
e^{-x}(1 + e^{2x}) = e^{-x} \cdot 1 + e^{-x} \cdot e^{2x}
\][/tex]
This simplifies to:
[tex]\[
e^{-x} + e^{-x} \cdot e^{2x}
\][/tex]
2. Combine Exponential Terms:
Simplify the term [tex]\(e^{-x} \cdot e^{2x}\)[/tex]:
[tex]\[
e^{-x} \cdot e^{2x} = e^{-x + 2x} = e^{x}
\][/tex]
So the integrand now becomes:
[tex]\[
e^{-x} + e^{x}
\][/tex]
3. Split the Integral:
Now, express the original integral as the sum of two simpler integrals:
[tex]\[
\int e^{-x}(1 + e^{2x}) \, dx = \int e^{-x} \, dx + \int e^{x} \, dx
\][/tex]
4. Integrate Each Term:
- For [tex]\(\int e^{-x} \, dx\)[/tex]:
Recall that the integral of [tex]\(e^{-x}\)[/tex] with respect to [tex]\(x\)[/tex] is [tex]\(-e^{-x}\)[/tex]:
[tex]\[
\int e^{-x} \, dx = -e^{-x}
\][/tex]
- For [tex]\(\int e^{x} \, dx\)[/tex]:
Recall that the integral of [tex]\(e^{x}\)[/tex] with respect to [tex]\(x\)[/tex] is [tex]\(e^{x}\)[/tex]:
[tex]\[
\int e^{x} \, dx = e^{x}
\][/tex]
5. Combine the Results:
Combine the results of both integrals:
[tex]\[
\int e^{-x} + \int e^{x} = -e^{-x} + e^{x}
\][/tex]
6. Include the Constant of Integration:
Don't forget the constant of integration [tex]\(C\)[/tex]:
[tex]\[
\int e^{-x}(1 + e^{2x}) \, dx = -e^{-x} + e^{x} + C
\][/tex]
Therefore, the final answer is:
[tex]\[
\boxed{-e^{-x} + e^{x} + C}
\][/tex]
