`int(e^(x)(x-2))/(x(x^(2)+e^(x)))dx AAx gt0` is equal to

Options

A. `ln(1+(e^(x))/(x^(2)))+c`
B. `ln(-(1)/(2)+(e^(x))/(x^(2)))+c`
C. `ln(2+(e^(x))/(x^(2)))+c`
D. `ln(x+(e^(x))/(x^(2)))+c`

Explanation

To solve the integral

\[ \int \frac{e^x (x - 2)}{x (x^2 + e^x)} \, dx, \]

we can follow these steps:

### Step 1: Rewrite the Integral
We start by rewriting the integral:

\[ \int \frac{e^x (x - 2)}{x (x^2 + e^x)} \, dx = \int \frac{e^x x - 2e^x}{x (x^2 + e^x)} \, dx. \]

### Step 2: Split the Integral
Next, we can split the integral into two parts:

\[ \int \frac{e^x x}{x (x^2 + e^x)} \, dx - 2 \int \frac{e^x}{x (x^2 + e^x)} \, dx. \]

This simplifies to:

\[ \int \frac{e^x}{x^2 + e^x} \, dx - 2 \int \frac{e^x}{x (x^2 + e^x)} \, dx. \]

### Step 3: Simplify the Denominator
We can rewrite the denominator \(x^2 + e^x\) and notice that we can add and subtract \(3x^2\) in the numerator:

\[ = \int \frac{e^x (3x^2 + e^x - 3x^2)}{x (x^2 + e^x)} \, dx. \]

### Step 4: Separate the Terms
Now we can separate the terms:

\[ \int \frac{3x^2 + e^x}{x^2 + e^x} \, dx - 3 \int \frac{x^2 + e^x}{x (x^2 + e^x)} \, dx. \]

### Step 5: Use Substitution
Let \(t = x^3 + x e^x\). Then, the differential \(dt\) can be expressed as:

\[ dt = (3x^2 + e^x) \, dx. \]

Thus, we can rewrite the integral as:

\[ \int \frac{dt}{t} - 3 \int \frac{dx}{x}. \]

### Step 6: Integrate
Now we can integrate both parts:

\[ \int \frac{dt}{t} = \ln |t| + C_1, \]
and
\[ -3 \int \frac{dx}{x} = -3 \ln |x| + C_2. \]

### Step 7: Combine the Results
Combining these results gives us:

\[ \ln |t| - 3 \ln |x| + C = \ln \left( \frac{t}{x^3} \right) + C. \]

### Step 8: Substitute Back for \(t\)
Substituting back for \(t\):

\[ \ln \left( \frac{x^3 + x e^x}{x^3} \right) + C = \ln \left( 1 + \frac{e^x}{x^2} \right) + C. \]

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

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