`intxe^(2x)(1+x)dx` equals

Options

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

Answer

To solve the integral \( \int x e^{2x} (1+x) \, dx \), we will use integration by parts and substitution. Here’s a step-by-step solution:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int x e^{2x} (1+x) \, dx \]
This can be expanded as:
\[ I = \int x e^{2x} \, dx + \int x^2 e^{2x} \, dx \]

### Step 2: Solve the First Integral
Let’s first solve \( \int x e^{2x} \, dx \) using integration by parts. We choose:
- \( u = x \) (thus \( du = dx \))
- \( dv = e^{2x} \, dx \) (thus \( v = \frac{1}{2} e^{2x} \))

Using integration by parts formula \( \int u \, dv = uv - \int v \, du \):
\[ \int x e^{2x} \, dx = x \cdot \frac{1}{2} e^{2x} - \int \frac{1}{2} e^{2x} \, dx \]
Now, we calculate the remaining integral:
\[ \int e^{2x} \, dx = \frac{1}{2} e^{2x} \]
Thus,
\[ \int x e^{2x} \, dx = \frac{1}{2} x e^{2x} - \frac{1}{4} e^{2x} \]

### Step 3: Solve the Second Integral
Next, we solve \( \int x^2 e^{2x} \, dx \) again using integration by parts. We choose:
- \( u = x^2 \) (thus \( du = 2x \, dx \))
- \( dv = e^{2x} \, dx \) (thus \( v = \frac{1}{2} e^{2x} \))

Using integration by parts:
\[ \int x^2 e^{2x} \, dx = x^2 \cdot \frac{1}{2} e^{2x} - \int \frac{1}{2} e^{2x} \cdot 2x \, dx \]
This simplifies to:
\[ \int x^2 e^{2x} \, dx = \frac{1}{2} x^2 e^{2x} - \int x e^{2x} \, dx \]
We already computed \( \int x e^{2x} \, dx \):
\[ \int x^2 e^{2x} \, dx = \frac{1}{2} x^2 e^{2x} - \left( \frac{1}{2} x e^{2x} - \frac{1}{4} e^{2x} \right) \]
Combining these results gives:
\[ \int x^2 e^{2x} \, dx = \frac{1}{2} x^2 e^{2x} - \frac{1}{2} x e^{2x} + \frac{1}{4} e^{2x} \]

### Step 4: Combine the Results
Now, we combine the results from both integrals:
\[ I = \left( \frac{1}{2} x e^{2x} - \frac{1}{4} e^{2x} \right) + \left( \frac{1}{2} x^2 e^{2x} - \frac{1}{2} x e^{2x} + \frac{1}{4} e^{2x} \right) \]
Simplifying this gives:
\[ I = \frac{1}{2} x^2 e^{2x} - \frac{1}{4} e^{2x} + \frac{1}{4} e^{2x} \]
Thus, the final result is:
\[ I = \frac{1}{2} x^2 e^{2x} + C \]

### Final Answer
\[ \int x e^{2x} (1+x) \, dx = \frac{1}{2} x^2 e^{2x} + C \]