Integration by Parts

This lesson covers integration by parts, a technique for integrating products of two functions. It is the reverse of the product rule for differentiation and is essential for A-Level Mathematics.

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The Formula

Integration by parts states:

∫ u (dv/dx) dx = uv − ∫ v (du/dx) dx

Or in the shorter notation:

∫ u dv = uv − ∫ v du

This formula converts one integral into another that is hopefully simpler.

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Derivation from the Product Rule

The product rule states:

d/dx [uv] = u(dv/dx) + v(du/dx)

Integrating both sides:

uv = ∫ u(dv/dx) dx + ∫ v(du/dx) dx

Rearranging:

∫ u(dv/dx) dx = uv − ∫ v(du/dx) dx

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Choosing u and dv/dx

The key to successful integration by parts is choosing which factor is u and which is dv/dx. The goal is to make ∫ v(du/dx) dx simpler than the original integral.

A useful guideline (sometimes called LIATE) is to let u be the function that comes first in this list: