`inte^(x)((1)/(x^(2))-(2)/(x^(3)))dx=?`

Options

A. `(-e^(x))/(x^(2))+C`
B. `(e^(x))/(x^(2))+C`
C. `e^(x)((-1)/(x)+(1)/(x^(2)))+C`
D. none of these

Explanation

To solve the integral \( I = \int e^x \left( \frac{1}{x^2} - \frac{2}{x^3} \right) dx \), we will use substitution and integration techniques. Let's go through the steps.

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int e^x \left( \frac{1}{x^2} - \frac{2}{x^3} \right) dx \]
This can be rewritten as:
\[ I = \int e^x \left( \frac{1}{x^2} \right) dx - 2 \int e^x \left( \frac{1}{x^3} \right) dx \]

### Step 2: Use Substitution
Let’s use the substitution:
\[ t = e^x \cdot \frac{1}{x^2} \]
Now, we need to differentiate \( t \) with respect to \( x \) to find \( dt \).

### Step 3: Differentiate Using the Product Rule
Using the product rule, we have:
\[ \frac{dt}{dx} = \frac{d}{dx} \left( e^x \cdot \frac{1}{x^2} \right) = e^x \cdot \frac{d}{dx} \left( \frac{1}{x^2} \right) + \frac{1}{x^2} \cdot \frac{d}{dx} \left( e^x \right) \]
Calculating the derivatives:
\[ \frac{d}{dx} \left( \frac{1}{x^2} \right) = -\frac{2}{x^3} \]
and
\[ \frac{d}{dx} \left( e^x \right) = e^x \]
Thus,
\[ \frac{dt}{dx} = e^x \left( -\frac{2}{x^3} \right) + e^x \cdot \frac{1}{x^2} = e^x \left( \frac{1}{x^2} - \frac{2}{x^3} \right) \]

### Step 4: Rearranging for \( dt \)
Now, we can express \( dt \) as:
\[ dt = e^x \left( \frac{1}{x^2} - \frac{2}{x^3} \right) dx \]
This means:
\[ I = \int dt \]

### Step 5: Integrate
Integrating both sides gives:
\[ I = t + C \]

### Step 6: Substitute Back
Recall that we defined \( t = e^x \cdot \frac{1}{x^2} \), so:
\[ I = e^x \cdot \frac{1}{x^2} + C \]

### Final Answer
Thus, the final result is:
\[ I = \frac{e^x}{x^2} + C \]