Read the following text and answer the followig questions on the basis of the same :
`inte^(x)[f(x) + f'(x)]dx = int e^(x)f(x)dx + int e^(x) f'(x)dx`
`= f(x)e^(x) - int f'(x)e^(x)dx + int f'(x)e^(x)dx`
`= e^(x)f(x) + c`
`int e^(x)((x-1)/(x^(2)))dx` = __________.

Options

A. `e^(x) + c`s
B. `(e^(x))/(x) + c`
C. `(e^(x))/(x^(2)) + c`
D. `(-e^(x))/(x^(2)) + c`

Answer

To solve the integral \( \int e^x \left( \frac{x-1}{x^2} \right) dx \), we will use integration by parts and the properties of exponential functions.

### Step-by-Step Solution:

1. **Rewrite the Integral:**
\[ \int e^x \left( \frac{x-1}{x^2} \right) dx = \int e^x \left( \frac{x}{x^2} - \frac{1}{x^2} \right) dx = \int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) dx \]

2. **Separate the Integral:**
\[ \int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) dx = \int e^x \cdot \frac{1}{x} dx - \int e^x \cdot \frac{1}{x^2} dx \]

3. **Identify \( f(x) \) and \( f'(x) \):**
Let \( f(x) = \frac{1}{x} \) and \( f'(x) = -\frac{1}{x^2} \).

4. **Use the Integration by Parts Formula:**
According to the formula given in the text:
\[ \int e^x [f(x) + f'(x)] dx = e^x f(x) + C \]
We can apply this to both integrals.

5. **Calculate Each Integral:**
- For \( \int e^x \cdot \frac{1}{x} dx \):
\[ e^x \cdot \frac{1}{x} + C_1 \]
- For \( \int e^x \cdot \frac{1}{x^2} dx \):
\[ e^x \cdot \left(-\frac{1}{x}\right) + C_2 \]

6. **Combine the Results:**
\[ \int e^x \cdot \frac{1}{x} dx - \int e^x \cdot \frac{1}{x^2} dx = e^x \cdot \frac{1}{x} + C_1 + e^x \cdot \frac{1}{x} + C_2 \]
This simplifies to:
\[ e^x \cdot \frac{1}{x} + C \]

7. **Final Result:**
Thus, the final answer is:
\[ \int e^x \left( \frac{x-1}{x^2} \right) dx = e^x \cdot \frac{1}{x} + C \]