Integrate x²/e^x dx

Answer 1

We can use integration by parts to solve this integral:

Let u = x² and dv/dx = e^(-x)dx. Then du/dx = 2x and v = -e^(-x).

Using the formula for integration by parts:

∫ u dv/dx dx = uv - ∫ v du/dx dx

Substituting in our values:

∫ x²/e^(x) dx = -x²e^(-x) - ∫ -2xe^(-x) dx

We can use integration by parts again to solve the second integral:

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

Using the formula for integration by parts again:

∫ u dv/dx dx = uv - ∫ v du/dx dx

Substituting in our values:

- ∫ -2xe^(-x) dx = 2xe^(-x) - ∫ -2e^(-x) dx

= 2xe^(-x) + 2e^(-x) + C, where C is the constant of integration.

Substituting this result back into our original equation:

∫ x²/e^(x) dx = -x²e^(-x) - (2xe^(-x) + 2e^(-x) + C)

= -(x² + 2x + 2)e^(-x) + C.

Therefore, the solution to the integral is ∫ x²/e^(x) dx = -(x² + 2x + 2)e^(-x) + C.