Question

How do you derive the formula for integration by parts?

Answer 1

The integration by parts formula is derived directly from the product rule for differentiability.

If `f` and `g` are continuously differentiable everywhere, then we can differentiate their product (using the product rule):

`d/dx (fg) = (f)( d/dx g) + (d/dx f)( g)`

`:. d/dx (fg) = f \ (dg)/dx + g \ (df)/dx`

`:. f \ (dg)/dx = d/dx (fg) - g \ (df)/dx`

Now simply integrate wrt `x`:

`int \ f \ (dg)/dx \ dx = int \ d/dx (fg) \ dx - int \ g \ (df)/dx \ dx`

From which we get the IBP formula:

`int \ f \ (dg)/dx \ dx = fg - int \ g \ (df)/dx \ dx`