How do you derive the formula for integration by parts?

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Answer 1 · 2017-05-30T16:15:50

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:

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