To evaluate the definite integral:

$\int_{0}^{1} xe^{-x^2} \, dx$

Solution:

Step 1: Substitution

Let $u = -x^2$. Then, $du = -2x \, dx \quad \Rightarrow \quad x \, dx = -\frac{1}{2} du$

Change the bounds accordingly:

• When $x = 0$, $u = 0$,
• When $x = 1$, $u = -1$.

The integral becomes: $\int_{0}^{1} xe^{-x^2} \, dx = \int_{0}^{-1} e^{u} \left(-\frac{1}{2}\right) du$

Step 2: Simplify and Change Bounds

Rewriting: $\int_{0}^{1} xe^{-x^2} \, dx = \frac{1}{2} \int_{-1}^{0} e^{u} \, du$

Step 3: Compute the Integral

The integral of $e^{u}$ is $e^{u}$: $\frac{1}{2} \int_{-1}^{0} e^{u} \, du = \frac{1}{2} \left[ e^{u} \right]_{-1}^{0}$

Evaluate the limits: [ \frac{1}{2} \left[ e^{0} - e^{-1} \right] = \frac{1}{2} \left[ 1 - \frac{1}{e} \right] ]

Simplify: $\frac{1}{2} \left( \frac{e - 1}{e} \right) = \frac{e - 1}{2e}$

Final Answer:

The value of the integral is: $\boxed{\frac{e - 1}{2e}}$

This corresponds to the last option.

Do you want further clarification or detailed steps on substitution and integration?

Related Questions:

1. How do you verify the substitution process in definite integrals?
2. What are the common integral properties when exponential functions are involved?
3. Can integration by parts be used for similar integrals, and how does it compare?
4. Why do we include bounds in substitution methods during definite integrals?
5. How does the exponential decay term $e^{-x^2}$ affect the integral's convergence?

Tip:

When substituting in definite integrals, always remember to adjust the bounds to match the new variable.