Integration by Parts using Tabular Approach xsin(x)

Integration by Parts using Tabular Approach

Given the integral:

∫ xsin(x) dx

We’ll use the tabular method to solve it:

1. Choose Functions:
• u = x
• dv = sin(x) 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:

u	dv
x	sin(x)
1	-cos(x)
0	-sin(x)
0	cos(x)

3. Apply Signs and Multiply Diagonally:

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

(+) × (xsin(x)) + (-) × (1 × -cos(x)) + (+) × (0 × -sin(x)) + (-) × (0 × cos(x))

Here’s how the multiplication works:

• First row: (+) × (xsin(x)) = xsin(x)
• Second row: (-) × (1 × -cos(x)) = cos(x)
• Third row: (+) × (0 × -sin(x)) = 0
• Fourth row: (-) × (0 × cos(x)) = 0
4. Final Result:

∫ xsin(x) dx = xsin(x) + cos(x) + C

where C is the constant of integration.