If `int_(0)^(1)e^(x^(2))(x-a)dx=0`, then the value of `int_(0)^(1)e^(X^(2))dx` is euqal to

Options

A. `(1)/(2a)(e-1)`
B. `(a)/(2)(e-1)`
C. `(1)/(2a)(e+1)`
D. `(a)/(2)(e+1)`

Answer

To solve the given problem, we need to evaluate the integral \( \int_{0}^{1} e^{x^2} dx \) given that \( \int_{0}^{1} e^{x^2} (x - a) dx = 0 \).

### Step-by-Step Solution:

1. **Set Up the Integral**:
We start with the equation:
\[ \int_{0}^{1} e^{x^2} (x - a) dx = 0 \]

2. **Expand the Integral**:
We can expand the integral:
\[ \int_{0}^{1} e^{x^2} x \, dx - a \int_{0}^{1} e^{x^2} \, dx = 0 \]

3. **Rearranging the Equation**:
Rearranging gives us:
\[ \int_{0}^{1} e^{x^2} x \, dx = a \int_{0}^{1} e^{x^2} \, dx \]

4. **Isolate the Integral**:
From the above equation, we can express \( a \) in terms of the integrals:
\[ a = \frac{\int_{0}^{1} e^{x^2} x \, dx}{\int_{0}^{1} e^{x^2} \, dx} \]

5. **Evaluate the Integral \( \int_{0}^{1} e^{x^2} x \, dx \)**:
To evaluate \( \int_{0}^{1} e^{x^2} x \, dx \), we can use substitution. Let \( t = x^2 \), then \( dt = 2x \, dx \) or \( x \, dx = \frac{dt}{2} \).
Changing the limits:
- When \( x = 0 \), \( t = 0 \)
- When \( x = 1 \), \( t = 1 \)

Thus, the integral becomes:
\[ \int_{0}^{1} e^{x^2} x \, dx = \int_{0}^{1} e^{t} \frac{dt}{2} = \frac{1}{2} \int_{0}^{1} e^{t} \, dt \]

6. **Evaluate \( \int_{0}^{1} e^{t} \, dt \)**:
The integral \( \int_{0}^{1} e^{t} \, dt \) evaluates to:
\[ e^{t} \bigg|_{0}^{1} = e - 1 \]
Therefore,
\[ \int_{0}^{1} e^{x^2} x \, dx = \frac{1}{2} (e - 1) \]

7. **Substituting Back**:
Now substituting back into our equation for \( a \):
\[ a = \frac{\frac{1}{2} (e - 1)}{\int_{0}^{1} e^{x^2} \, dx} \]

8. **Finding \( \int_{0}^{1} e^{x^2} \, dx \)**:
We now have:
\[ \int_{0}^{1} e^{x^2} \, dx = \frac{\frac{1}{2} (e - 1)}{a} \]

9. **Final Expression**:
Therefore, we can express the integral \( \int_{0}^{1} e^{x^2} \, dx \) in terms of \( a \):
\[ \int_{0}^{1} e^{x^2} \, dx = \frac{e - 1}{2a} \]

### Conclusion:
The value of \( \int_{0}^{1} e^{x^2} \, dx \) is given by:
\[ \int_{0}^{1} e^{x^2} \, dx = \frac{e - 1}{2a} \]