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An identity relating the integral of a product to boundary terms and another integral.
Integration by parts: Let and let be functions that are differentiable on , with and Riemann integrable on . Then
This is the integral form of the product rule from differentiation rules, typically justified using fundamental theorem of calculus II.
A differentiable function (one variable) is a function that has a derivative at a point (meaning exists), and it is differentiable on if it is differentiable at every point of .
Differentiation rules: Let be an interval, and let be differentiable at a point . Then:
Fundamental theorem of calculus II: Let and let be a Riemann integrable function. Suppose is continuous on , differentiable on , and satisfies for all . Then
Combined with fundamental theorem of calculus I, this explains why differentiation rules can be used to evaluate definite integrals via antiderivatives.