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 \( g(-1) \).

### Step-by-Step Solution:

1. **Rewrite the Integral**:
We start with the integral:
\[ \int x^5 e^{-x^2} \, dx \]
To simplify the integration, we can use the substitution \( t = x^2 \). Then, \( dt = 2x \, dx \) or \( dx = \frac{dt}{2\sqrt{t}} \).

2. **Change of Variables**:
Substitute \( x^2 = t \) into the integral:
\[ \int x^5 e^{-x^2} \, dx = \int (t^{5/2}) e^{-t} \frac{dt}{2\sqrt{t}} = \frac{1}{2} \int t^2 e^{-t} \, dt \]

3. **Integration by Parts**:
Now, we will use integration by parts on \( \int t^2 e^{-t} \, dt \). Let:
- \( u = t^2 \) and \( dv = e^{-t} dt \)
- Then, \( du = 2t \, dt \) and \( v = -e^{-t} \)

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

4. **Second Integration by Parts**:
We need to integrate \( \int 2t e^{-t} \, dt \) again by parts:
- Let \( u = 2t \) and \( dv = e^{-t} dt \)
- Then, \( du = 2 \, dt \) and \( v = -e^{-t} \)

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

5. **Combine Results**:
Now substituting back:
\[ \int t^2 e^{-t} \, dt = -t^2 e^{-t} - 2t e^{-t} - 2 e^{-t} \]
Thus,
\[ \int t^2 e^{-t} \, dt = -e^{-t}(t^2 + 2t + 2) \]

6. **Substituting Back**:
Now substituting back for \( t = x^2 \):
\[ \int x^5 e^{-x^2} \, dx = \frac{1}{2} \left(-e^{-x^2}(x^4 + 2x^2 + 2)\right) + C \]
Simplifying gives:
\[ \int x^5 e^{-x^2} \, dx = -\frac{1}{2} e^{-x^2}(x^4 + 2x^2 + 2) + C \]

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

8. **Finding \( g(-1) \)**:
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, \( g(-1) = -\frac{5}{2} \).