Find the integral of xeˣ using the tabular method

Integration by Parts using Tabular Approach

Let’s find the integral of xeˣ using the tabular method:

1. 1. Choose Functions:
      • u = x
      • dv = eˣ dx
   2. Create a Table:

For the table, alternate between differentiating u and integrating dv. Start by listing u, its successive derivatives, and dv, its successive integrals:

1. 1. Apply Signs and Multiply Diagonally:

Now, apply alternating signs down the table. Multiply the terms diagonally and add them up:

(+) × (xeˣ) + (-) × (eˣ) + (+) × (eˣ)

Here’s how the multiplication works:

1. 1. • First row: (+) × (xeˣ) = xeˣ
      • Second row: (-) × (eˣ) = -eˣ
      • Third row: (+) × (eˣ) = eˣ
   2. Final Result:

The integral of xeˣ is:

∫ xeˣ dx = xeˣ – eˣ + C

where C is the constant of integration.