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. `-(1)/(2)`
D. `-(5)/(2)`

Explanation

To solve the integral \( \int x^5 e^{-x^2} \, dx \) and express it in the form \( g(x)e^{-x^2} + C \), we will use integration by parts and substitution. Here’s a step-by-step solution:

### Step 1: Substitution
Let \( t = -x^2 \). Then, we have:
\[ dt = -2x \, dx \quad \Rightarrow \quad dx = -\frac{1}{2x} \, dt \]
Also, \( x^2 = -t \) implies \( x^4 = (-t)^2 = t^2 \).

### Step 2: Rewrite the Integral
Substituting \( t \) into the integral:
\[ \int x^5 e^{-x^2} \, dx = \int x^5 e^t \left(-\frac{1}{2x}\right) dt = -\frac{1}{2} \int x^4 e^t \, dt \]
Now, substituting \( x^4 = t^2 \):
\[ -\frac{1}{2} \int t^2 e^t \, dt \]

### Step 3: Integration by Parts
Using integration by parts, let:
- \( u = t^2 \) and \( dv = e^t \, dt \)
- Then, \( du = 2t \, dt \) and \( v = e^t \)

Applying integration by parts:
\[ \int t^2 e^t \, dt = t^2 e^t - \int 2t e^t \, dt \]

### Step 4: Second Integration by Parts
For the integral \( \int 2t e^t \, dt \):
- Let \( u = 2t \) and \( dv = e^t \, dt \)
- Then, \( du = 2 \, dt \) and \( v = e^t \)

Thus,
\[ \int 2t e^t \, dt = 2t e^t - \int 2 e^t \, dt = 2t e^t - 2e^t \]

### Step 5: Combine Results
Substituting back:
\[ \int t^2 e^t \, dt = t^2 e^t - (2t e^t - 2e^t) = t^2 e^t - 2t e^t + 2e^t \]
So,
\[ -\frac{1}{2} \int t^2 e^t \, dt = -\frac{1}{2} \left( t^2 e^t - 2t e^t + 2e^t \right) \]

### Step 6: Substitute Back to \( x \)
Now, substituting back \( t = -x^2 \):
\[ = -\frac{1}{2} \left( (-x^2)^2 e^{-x^2} - 2(-x^2)e^{-x^2} + 2e^{-x^2} \right) \]
This simplifies to:
\[ = -\frac{1}{2} \left( x^4 e^{-x^2} + 2x^2 e^{-x^2} + 2e^{-x^2} \right) \]
Thus,
\[ = -\frac{1}{2} x^4 e^{-x^2} - x^2 e^{-x^2} - e^{-x^2} + C \]

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

### Step 8: Find \( g(-1) \)
Now, substituting \( x = -1 \):
\[ g(-1) = -\frac{1}{2} (-1)^4 - (-1)^2 - 1 = -\frac{1}{2}(1) - 1 - 1 = -\frac{1}{2} - 1 - 1 = -\frac{1}{2} - 2 = -\frac{5}{2} \]

### Final Answer
Thus, \( g(-1) = -\frac{5}{2} \).

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