If `int x^(5)e^(-x^(2))dx = g(x)e^(-x^(2))+C`, where C is a constant of integration, then g(-1) is equal to

Options

A. `1 `
B. ` - 1 `
C. ` - (5 )/(2)`
D. ` -(1)/(2) `

Answer

To solve the integral \( \int x^5 e^{-x^2} \, dx = g(x) e^{-x^2} + C \), where \( C \) is a constant of integration, we need to find the function \( g(x) \) and then evaluate \( g(-1) \).

### Step 1: Rewrite the Integral
We start with the integral:
\[ \int x^5 e^{-x^2} \, dx \]
We can rewrite \( x^5 \) as \( x \cdot x^4 \):
\[ \int x^5 e^{-x^2} \, dx = \int x \cdot x^4 e^{-x^2} \, dx \]

### Step 2: Use Substitution
Let \( t = -x^2 \). Then, \( dt = -2x \, dx \) or \( dx = \frac{dt}{-2x} \). This gives us:
\[ x^4 = (-t)^2 = t^2 \]
Thus, the integral becomes:
\[ \int x^5 e^{-x^2} \, dx = \int x \cdot t^2 e^{t} \cdot \frac{dt}{-2x} = -\frac{1}{2} \int t^2 e^{t} \, dt \]

### Step 3: Integrate by Parts
Now we need to integrate \( \int t^2 e^{t} \, dt \) using integration by parts. Let:
- \( u = t^2 \) and \( dv = e^{t} dt \)
Then:
- \( du = 2t \, dt \)
- \( v = e^{t} \)

Using integration by parts:
\[ \int u \, dv = uv - \int v \, du \]
We have:
\[ \int t^2 e^{t} \, dt = t^2 e^{t} - \int 2t e^{t} \, dt \]

### Step 4: Integrate Again by Parts
Now we need to integrate \( \int 2t e^{t} \, dt \) again by parts:
Let:
- \( u = 2t \) and \( dv = e^{t} dt \)
Then:
- \( du = 2 \, dt \)
- \( v = e^{t} \)

So:
\[ \int 2t e^{t} \, dt = 2t e^{t} - \int 2 e^{t} \, dt = 2t e^{t} - 2 e^{t} \]

### Step 5: Substitute Back
Now substitute back into our integral:
\[ \int t^2 e^{t} \, dt = t^2 e^{t} - (2t e^{t} - 2 e^{t}) = e^{t}(t^2 - 2t + 2) \]

### Step 6: Substitute \( t = -x^2 \)
Substituting back \( t = -x^2 \):
\[ \int x^5 e^{-x^2} \, dx = -\frac{1}{2} e^{-x^2}((-x^2)^2 - 2(-x^2) + 2) + C \]
This simplifies to:
\[ = -\frac{1}{2} e^{-x^2}(x^4 + 2x^2 + 2) + C \]

### Step 7: Identify \( g(x) \)
From the equation \( \int x^5 e^{-x^2} \, dx = g(x) e^{-x^2} + C \), we can identify:
\[ g(x) = -\frac{1}{2}(x^4 + 2x^2 + 2) \]

### Step 8: Evaluate \( g(-1) \)
Now, we need to find \( g(-1) \):
\[ g(-1) = -\frac{1}{2}((-1)^4 + 2(-1)^2 + 2) = -\frac{1}{2}(1 + 2 + 2) = -\frac{1}{2}(5) = -\frac{5}{2} \]

### Final Answer
Thus, the value of \( g(-1) \) is:
\[ \boxed{-\frac{5}{2}} \]