Evaluate the definite integrals`int_0^1x e^(x^2 )dx`

Explanation

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

### Step 1: Substitution
Let \( t = x^2 \). Then, we differentiate both sides to find \( dx \):
\[ dt = 2x \, dx \quad \Rightarrow \quad dx = \frac{dt}{2x} \]
Since \( t = x^2 \), we can express \( x \) in terms of \( t \):
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