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
# 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
# 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 #