If `int x^2e^(-2x)= e^(-2x)(ax^2+bx+c)+d` then

Options

A. `a= -(1)/(2)`
B. `b =(1)/(2)`
C. `c= -(1)/(4)`
D. `d in R`

Explanation

To solve the integral \( \int x^2 e^{-2x} \, dx \) and express it in the form \( e^{-2x}(ax^2 + bx + c) + d \), we will use the method of integration by parts.

### Step 1: Set up for integration by parts
We will choose:
- \( u = x^2 \) (which we will differentiate)
- \( dv = e^{-2x} \, dx \) (which we will integrate)

### Step 2: Differentiate and integrate
Now we differentiate \( u \) and integrate \( dv \):
- \( du = 2x \, dx \)
- \( v = \int e^{-2x} \, dx = -\frac{1}{2} e^{-2x} \)

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

### Step 4: Solve the remaining integral \( \int x e^{-2x} \, dx \)
We will apply integration by parts again on \( \int x e^{-2x} \, dx \):
- Let \( u = x \) and \( dv = e^{-2x} \, dx \)
- Then \( du = dx \) and \( v = -\frac{1}{2} e^{-2x} \)

Applying integration by parts:
\[ \int x e^{-2x} \, dx = uv - \int v \, du \]
Substituting the values:
\[ = x \left(-\frac{1}{2} e^{-2x}\right) - \int \left(-\frac{1}{2} e^{-2x}\right) \, dx \]
This simplifies to:
\[ = -\frac{1}{2} x e^{-2x} + \frac{1}{4} e^{-2x} \]

### Step 5: Combine results
Now we substitute back into our original equation:
\[ \int x^2 e^{-2x} \, dx = -\frac{1}{2} x^2 e^{-2x} - \frac{1}{2} x e^{-2x} + \frac{1}{4} e^{-2x} + C \]
Factoring out \( e^{-2x} \):
\[ = e^{-2x} \left(-\frac{1}{2} x^2 - \frac{1}{2} x + \frac{1}{4}\right) + C \]

### Step 6: Identify coefficients
Now we can compare this with the given form \( e^{-2x}(ax^2 + bx + c) + d \):
- \( a = -\frac{1}{2} \)
- \( b = -\frac{1}{2} \)
- \( c = \frac{1}{4} \)
- \( d = C \) (a constant)

### Final Answer
Thus, the values of \( a \), \( b \), and \( c \) are:
- \( a = -\frac{1}{2} \)
- \( b = -\frac{1}{2} \)
- \( c = \frac{1}{4} \)