`inte^(x)((2)/(x)-(2)/(x^(2)))dx` is equal to

Options

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

Answer

To solve the integral \( \int e^x \left( \frac{2}{x} - \frac{2}{x^2} \right) dx \), we can follow these steps:

### Step 1: Factor out the constant
We can factor out the constant \(2\) from the integral:

\[ \int e^x \left( \frac{2}{x} - \frac{2}{x^2} \right) dx = 2 \int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) dx \]

**Hint:** Remember that constants can be factored out of integrals.

### Step 2: Rewrite the integrand
Now we can rewrite the integrand:

\[ \int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) dx = \int e^x \left( \frac{1}{x} + \left(-\frac{1}{x^2}\right) \right) dx \]

**Hint:** Look for a way to express the integrand as a function plus its derivative.

### Step 3: Identify \(f(x)\) and \(f'(x)\)
Let’s assume \(f(x) = \frac{1}{x}\). Then, we find its derivative:

\[ f'(x) = -\frac{1}{x^2} \]

Now we can see that:

\[ \frac{1}{x} - \frac{1}{x^2} = f(x) + f'(x) \]

**Hint:** Recognize the relationship between a function and its derivative to simplify the integration.

### Step 4: Apply the integration property
Using the property of integration, we have:

\[ \int e^x (f(x) + f'(x)) dx = e^x f(x) + C \]

Thus,

\[ \int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) dx = e^x \cdot \frac{1}{x} + C \]

**Hint:** Familiarize yourself with integration properties that involve exponential functions and their derivatives.

### Step 5: Multiply by the constant
Now, we multiply back by the constant we factored out earlier:

\[ 2 \int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) dx = 2 \left( e^x \cdot \frac{1}{x} + C \right) \]

This simplifies to:

\[ = \frac{2 e^x}{x} + C' \]

where \(C' = 2C\) is still a constant.

**Hint:** Always remember to include the constant of integration when you finish integrating.

### Final Answer
Thus, the final result of the integral is:

\[ \int e^x \left( \frac{2}{x} - \frac{2}{x^2} \right) dx = \frac{2 e^x}{x} + C \]