`int (x-2)/(x^(3)).e^(x) dx`

Answer

To solve the integral \( \int \frac{x - 2}{x^3} e^x \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We start by rewriting the integral in a more manageable form:
\[ I = \int \frac{x - 2}{x^3} e^x \, dx = \int \left( \frac{x}{x^3} - \frac{2}{x^3} \right) e^x \, dx = \int \left( \frac{1}{x^2} - \frac{2}{x^3} \right) e^x \, dx \]

### Step 2: Split the Integral
Now we can split the integral into two separate integrals:
\[ I = \int \frac{1}{x^2} e^x \, dx - 2 \int \frac{1}{x^3} e^x \, dx \]

### Step 3: Apply Integration by Parts
For the integral \( \int \frac{1}{x^2} e^x \, dx \), we can use integration by parts. Let:
- \( u = \frac{1}{x^2} \) (then \( du = -\frac{2}{x^3} \, dx \))
- \( dv = e^x \, dx \) (then \( v = e^x \))

Using the integration by parts formula \( \int u \, dv = uv - \int v \, du \):
\[ \int \frac{1}{x^2} e^x \, dx = \frac{1}{x^2} e^x - \int e^x \left(-\frac{2}{x^3}\right) \, dx \]
This simplifies to:
\[ \int \frac{1}{x^2} e^x \, dx = \frac{1}{x^2} e^x + 2 \int \frac{1}{x^3} e^x \, dx \]

### Step 4: Substitute Back
Now we can substitute this back into our original integral:
\[ I = \left( \frac{1}{x^2} e^x + 2 \int \frac{1}{x^3} e^x \, dx \right) - 2 \int \frac{1}{x^3} e^x \, dx \]
This gives us:
\[ I = \frac{1}{x^2} e^x + 2 \int \frac{1}{x^3} e^x \, dx - 2 \int \frac{1}{x^3} e^x \, dx \]
The \( 2 \int \frac{1}{x^3} e^x \, dx \) terms cancel out:
\[ I = \frac{1}{x^2} e^x \]

### Step 5: Final Result
Thus, the final result of the integral is:
\[ I = \frac{e^x}{x^2} + C \]