The integral is ∫x²eˣ dx = x² eˣ - 2xeˣ + 2eˣ + C
How to integrate using part integration.
Let's use integration by parts. The integration by parts formula is given by:
∫udv = uv - ∫vdu
Let's choose u = x² and dv = eˣd
du = 2xdx
v = ∫eˣ dx = eˣ
Now, apply the integration by parts formula:
∫x² eˣdx = x² eˣ - ∫2x eˣ dx
∫x² eˣdx = x² eˣ - ∫2x eˣdx
Let u = 2x and dv = eˣdx then calculate du and v.
du = 2dx
v = ∫eˣdx = eˣ
Apply the integration by parts formula again:
∫2x eˣdx = 2xeˣ - ∫2eˣdx
∫2eˣ dx = 2eˣ
∫x² eˣ dx = x² eˣ - (2xeˣ - 2eˣ)
Combine like terms:
∫x²eˣ dx = x² eˣ - 2xeˣ + 2eˣ + C
where C is the constant of integration.