The value of `int_0^1 e^(x^2) x dx` is equal to:

Options

A. `1/3(e-1)`
B. `1/2(e-1)`
C. `1/3(e+1)`
D. `1/2(e+1)`

Answer

To solve the integral \( I = \int_0^1 e^{x^2} x \, dx \), we will use a substitution method. Here are the steps:

### Step 1: Set up the integral
Let
\[ I = \int_0^1 e^{x^2} x \, dx \]

### Step 2: Choose a substitution
We will use the substitution \( t = x^2 \). Then, we differentiate to find \( dt \):
\[ dt = 2x \, dx \quad \Rightarrow \quad x \, dx = \frac{dt}{2} \]

### Step 3: Change the limits of integration
When \( x = 0 \):
\[ t = 0^2 = 0 \]
When \( x = 1 \):
\[ t = 1^2 = 1 \]
Thus, the limits of integration change from \( x = 0 \) to \( x = 1 \) into \( t = 0 \) to \( t = 1 \).

### Step 4: Substitute into the integral
Substituting \( t \) and \( x \, dx \) into the integral gives:
\[ I = \int_0^1 e^t \cdot \frac{dt}{2} \]
This simplifies to:
\[ I = \frac{1}{2} \int_0^1 e^t \, dt \]

### Step 5: Integrate \( e^t \)
The integral of \( e^t \) is:
\[ \int e^t \, dt = e^t + C \]
Thus,
\[ I = \frac{1}{2} \left[ e^t \right]_0^1 \]

### Step 6: Evaluate the definite integral
Calculating the definite integral:
\[ I = \frac{1}{2} \left( e^1 - e^0 \right) = \frac{1}{2} \left( e - 1 \right) \]

### Final Result
Thus, the value of the integral is:
\[ I = \frac{e - 1}{2} \]