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 problem, we need to evaluate the integral given the equation:

\[ \int_{0}^{1} e^{x^2} (x - a) \, dx = 0 \]

### Step 1: Rewrite the Integral
We can break down the integral into two parts:

\[ \int_{0}^{1} e^{x^2} x \, dx - a \int_{0}^{1} e^{x^2} \, dx = 0 \]

### Step 2: Define the Integrals
Let:

\[ I = \int_{0}^{1} e^{x^2} \, dx \]

Then the equation becomes:

\[ \int_{0}^{1} e^{x^2} x \, dx = a I \]

### Step 3: Change of Variables
To evaluate \(\int_{0}^{1} e^{x^2} x \, dx\), we can use a substitution. Let \(t = x^2\), then \(dt = 2x \, dx\) or \(dx = \frac{dt}{2\sqrt{t}}\).

When \(x = 0\), \(t = 0\) and when \(x = 1\), \(t = 1\). Thus, we can rewrite the integral as:

\[ \int_{0}^{1} e^{x^2} x \, dx = \int_{0}^{1} e^t \frac{dt}{2} \]

### Step 4: Evaluate the Integral
Now, we evaluate the integral:

\[ \int_{0}^{1} e^t \, dt = e^t \Big|_{0}^{1} = e - 1 \]

Thus,

\[ \int_{0}^{1} e^{x^2} x \, dx = \frac{1}{2} (e - 1) \]

### Step 5: Substitute Back
Substituting back into our equation gives:

\[ \frac{1}{2} (e - 1) = a I \]

### Step 6: Solve for \(I\)
Now we can express \(I\) in terms of \(a\):

\[ I = \frac{1}{2a} (e - 1) \]

### Conclusion
Thus, the value of the integral \(\int_{0}^{1} e^{x^2} \, dx\) is:

\[ \int_{0}^{1} e^{x^2} \, dx = \frac{1}{2a} (e - 1) \]