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This lesson covers integration by parts — a technique for integrating the product of two functions — as required by the Edexcel A-Level Mathematics specification (9MA0). You need to know the formula, how to choose u and dv/dx, when to apply the method repeatedly, and the LIATE rule for guidance.
Integration by parts is derived from the product rule for differentiation. The formula is:
∫ u (dv/dx) dx = uv - ∫ v (du/dx) dx
Alternatively, using the shorthand notation:
∫ u dv = uv - ∫ v du
The product rule states: d/dx(uv) = u(dv/dx) + v(du/dx)
Rearranging: u(dv/dx) = d/dx(uv) - v(du/dx)
Integrating both sides: ∫ u(dv/dx) dx = uv - ∫ v(du/dx) dx
The key to integration by parts is choosing the right function for u and the right function for dv/dx. The goal is to make the resulting integral ∫ v(du/dx) dx simpler than the original.
A useful guideline is the LIATE rule. Choose u to be the function that comes first in this list:
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