Evaluate `int e^(x) cos^(2) x dx`

Answer

To evaluate the integral \( I = \int e^x \cos^2 x \, dx \), we will use the method of integration by parts.

### Step-by-Step Solution:

1. **Identify parts for integration by parts**:
We will let:
- \( u = \cos^2 x \) (first function)
- \( dv = e^x \, dx \) (second function)

2. **Differentiate and integrate**:
- Differentiate \( u \):
\[ du = -2 \cos x \sin x \, dx = -\sin(2x) \, dx \]
- Integrate \( dv \):
\[ v = \int e^x \, dx = e^x \]

3. **Apply the integration by parts formula**:
The integration by parts formula is:
\[ \int u \, dv = uv - \int v \, du \]
Applying this, we have:
\[ I = \cos^2 x \cdot e^x - \int e^x (-\sin(2x)) \, dx \]
Simplifying, we get:
\[ I = e^x \cos^2 x + \int e^x \sin(2x) \, dx \]

4. **Evaluate the new integral**:
Let’s denote the new integral as \( I_1 = \int e^x \sin(2x) \, dx \). We will again use integration by parts:
- Let:
- \( u = \sin(2x) \)
- \( dv = e^x \, dx \)
- Then:
- \( du = 2 \cos(2x) \, dx \)
- \( v = e^x \)

5. **Apply integration by parts again**:
\[ I_1 = \sin(2x) e^x - \int e^x (2 \cos(2x)) \, dx \]
Simplifying, we have:
\[ I_1 = e^x \sin(2x) - 2 \int e^x \cos(2x) \, dx \]

6. **Denote the new integral**:
Let \( I_2 = \int e^x \cos(2x) \, dx \). We will again apply integration by parts:
- Let:
- \( u = \cos(2x) \)
- \( dv = e^x \, dx \)
- Then:
- \( du = -2 \sin(2x) \, dx \)
- \( v = e^x \)

7. **Apply integration by parts**:
\[ I_2 = \cos(2x) e^x - \int e^x (-2 \sin(2x)) \, dx \]
Simplifying, we have:
\[ I_2 = e^x \cos(2x) + 2 I_1 \]

8. **Combine the equations**:
Now substituting \( I_2 \) back into the equation for \( I_1 \):
\[ I_1 = e^x \sin(2x) - 2(e^x \cos(2x) + 2 I_1) \]
Rearranging gives:
\[ I_1 + 4 I_1 = e^x \sin(2x) - 2 e^x \cos(2x) \]
Thus:
\[ 5 I_1 = e^x (\sin(2x) - 2 \cos(2x)) \]
Therefore:
\[ I_1 = \frac{e^x (\sin(2x) - 2 \cos(2x))}{5} \]

9. **Substituting back to find \( I \)**:
Now substituting \( I_1 \) back into the equation for \( I \):
\[ I = e^x \cos^2 x + \frac{e^x (\sin(2x) - 2 \cos(2x))}{5} \]

10. **Final result**:
The final result for the integral is:
\[ I = e^x \left( \cos^2 x + \frac{\sin(2x) - 2 \cos(2x)}{5} \right) + C \]