`inte^x (x)/((x+2)^3)dx =`

Options

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

Answer

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

### Step 1: Rewrite the Integrand
We start by rewriting the integrand:
\[ \frac{e^x \cdot x}{(x+2)^3} = \frac{e^x \cdot (x + 2 - 2)}{(x+2)^3} = \frac{e^x \cdot (x + 2)}{(x+2)^3} - \frac{2e^x}{(x+2)^3} \]
This simplifies to:
\[ \frac{e^x}{(x+2)^2} - \frac{2e^x}{(x+2)^3} \]

### Step 2: Split the Integral
Now we can split the integral into two parts:
\[ \int \frac{e^x}{(x+2)^2} \, dx - 2 \int \frac{e^x}{(x+2)^3} \, dx \]

### Step 3: Apply Integration by Parts
For the first integral \( \int \frac{e^x}{(x+2)^2} \, dx \), we will use integration by parts. Let:
- \( u = \frac{1}{(x+2)^2} \) (algebraic part)
- \( dv = e^x \, dx \) (exponential part)

Then we differentiate and integrate:
- \( du = -\frac{2}{(x+2)^3} \, dx \)
- \( v = e^x \)

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

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

### Step 5: Final Answer
Thus, the final answer is:
\[ \int \frac{e^x \cdot x}{(x+2)^3} \, dx = \frac{e^x}{(x+2)^2} + C \]