Integrate. (X^2) exponential (-x^2) dx ? Using limits as 0 and 2?

To integrate (x^2) * e^(-x^2) dx, we can use integration by parts.

Let u = x^2 and dv = e^(-x^2) dx
Then, du = 2x dx and v = -1/2 * e^(-x^2)

Using the integration by parts formula:
∫u dv = uv - ∫v du

∫(x^2) * e^(-x^2) dx = -1/2 * x^2 * e^(-x^2) - ∫(-1/2 * e^(-x^2) * 2x dx)
= -1/2 * x^2 * e^(-x^2) + ∫x * e^(-x^2) dx

Now, we can use substitution method to integrate the remaining integral:
Let t = -x^2
Then, dt = -2x dx
dx = -1/2 * dt

Substitute t = -x^2 and dx = -1/2 * dt into the integral:
∫x * e^(-x^2) dx = ∫-1/2 * e^t dt
= -1/2 * e^t + C
= -1/2 * e^(-x^2) + C

Therefore, the final result is:
∫(x^2) * e^(-x^2) dx = -1/2 * x^2 * e^(-x^2) - 1/2 * e^(-x^2) + C