Evaluate the following integration
`int 2^(x)*e^(x)*dx`

Answer

To evaluate the integral \( I = \int 2^x e^x \, dx \), we can use the integration by parts method, specifically the ILATE rule, which helps us choose which function to differentiate and which to integrate.

### Step-by-Step Solution:

1. **Identify Functions**:
We will let:
- \( u = 2^x \) (an exponential function)
- \( dv = e^x \, dx \)

2. **Differentiate and Integrate**:
Now we need to find \( du \) and \( v \):
- Differentiate \( u \):
\[ du = \frac{d}{dx}(2^x) \, dx = 2^x \ln(2) \, dx \]
- Integrate \( dv \):
\[ v = \int e^x \, dx = e^x \]

3. **Apply Integration by Parts**:
According to the integration by parts formula:
\[ \int u \, dv = uv - \int v \, du \]
Substituting our values:
\[ I = 2^x e^x - \int e^x (2^x \ln(2)) \, dx \]
This simplifies to:
\[ I = 2^x e^x - \ln(2) \int 2^x e^x \, dx \]
Notice that the integral on the right is the same as \( I \).

4. **Rearranging the Equation**:
We can express this as:
\[ I + \ln(2) I = 2^x e^x \]
Factoring out \( I \):
\[ I(1 + \ln(2)) = 2^x e^x \]

5. **Solve for \( I \)**:
Finally, we can solve for \( I \):
\[ I = \frac{2^x e^x}{1 + \ln(2)} + C \]
where \( C \) is the constant of integration.

### Final Answer:
\[ \int 2^x e^x \, dx = \frac{2^x e^x}{1 + \ln(2)} + C \]