How do you derive the formula for integration by parts?

1 Answer
May 30, 2017

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 #