`int(xe^(x))/((x+2)^(3))dx=`

Options

A. `(e^(x))/(x+2)+c`
B. `(e^(x))/((x+2)^(2))+c`
C. `(e^(x))/((x+2)^(3))+c`
D. `(xe^(x))/((x+2)^(2))+c`

Answer

To solve the integral \( \int \frac{x e^x}{(x+2)^3} \, dx \), we can use integration techniques involving substitution and integration by parts. Here’s a step-by-step solution:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int \frac{x e^x}{(x+2)^3} \, dx \]
We can rewrite \( x \) as \( (x + 2) - 2 \):
\[ I = \int \frac{((x + 2) - 2)e^x}{(x + 2)^3} \, dx \]
This simplifies to:
\[ I = \int \frac{(x + 2)e^x}{(x + 2)^3} \, dx - 2 \int \frac{e^x}{(x + 2)^3} \, dx \]
\[ I = \int \frac{e^x}{(x + 2)^2} \, dx - 2 \int \frac{e^x}{(x + 2)^3} \, dx \]

### Step 2: Solve the First Integral
Let’s denote:
\[ I_1 = \int \frac{e^x}{(x + 2)^2} \, dx \]
Using the integration property \( \int e^x f(x) \, dx = e^x f(x) - \int e^x f'(x) \, dx \), we can set \( f(x) = \frac{1}{(x + 2)^2} \).

Now, we find \( f'(x) \):
\[ f'(x) = -\frac{2}{(x + 2)^3} \]
Thus, we have:
\[ I_1 = e^x \cdot \frac{1}{(x + 2)^2} - \int e^x \left(-\frac{2}{(x + 2)^3}\right) \, dx \]
\[ I_1 = \frac{e^x}{(x + 2)^2} + 2 \int \frac{e^x}{(x + 2)^3} \, dx \]

### Step 3: Substitute Back into the Original Integral
Now, substituting \( I_1 \) back into the equation for \( I \):
\[ I = I_1 - 2 \int \frac{e^x}{(x + 2)^3} \, dx \]
\[ I = \left( \frac{e^x}{(x + 2)^2} + 2 \int \frac{e^x}{(x + 2)^3} \, dx \right) - 2 \int \frac{e^x}{(x + 2)^3} \, dx \]
\[ I = \frac{e^x}{(x + 2)^2} \]

### Step 4: Final Answer
Thus, the final result for the integral is:
\[ \int \frac{x e^x}{(x + 2)^3} \, dx = \frac{e^x}{(x + 2)^2} + C \]