What is the value of `int_(0)^(1) xe^(x^(2)) dx `?

Options

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

Explanation

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

### Step 1: Choose a substitution
Let \( t = x^2 \). Then, we differentiate to find \( dt \):
\[ dt = 2x \, dx \quad \Rightarrow \quad dx = \frac{dt}{2x} \]
From our substitution, we can express \( x \) in terms of \( t \):
\[ x = \sqrt{t} \]

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

### Step 3: Substitute into the integral
Now we substitute \( x \) and \( dx \) into the integral:
\[ \int_0^1 x e^{x^2} \, dx = \int_0^1 \sqrt{t} e^t \cdot \frac{dt}{2\sqrt{t}} = \int_0^1 \frac{1}{2} e^t \, dt \]

### Step 4: Simplify the integral
This simplifies to:
\[ \frac{1}{2} \int_0^1 e^t \, dt \]

### Step 5: Evaluate the integral
The integral of \( e^t \) is \( e^t \), so we evaluate:
\[ \int_0^1 e^t \, dt = e^1 - e^0 = e - 1 \]

### Step 6: Multiply by the constant
Now we multiply by \( \frac{1}{2} \):
\[ \frac{1}{2} (e - 1) \]

### Final Answer
Thus, the value of the integral \( \int_0^1 x e^{x^2} \, dx \) is:
\[ \frac{e - 1}{2} \]