The idea

Integration by parts

Reversing the product rule to integrate a product — ∫u(dv/dx)dx = uv − ∫v(du/dx)dx — choosing u to differentiate to something simpler, the ∫ln x trick, and applying it twice.

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Pure · Integration

Integration by parts

Reversing the product rule to integrate a product — ∫u(dv/dx)dx = uv − ∫v(du/dx)dx — choosing u to differentiate to something simpler, the ∫ln x trick, and applying it twice.

Why it works

The product rule, reversed

Some products can't be integrated by recognition —

\int x e^x\,dx

has no inside function whose derivative is sitting there. The reverse chain rule needs a function-of-a-function with its derivative alongside;

x e^x

is a plain product of two unrelated factors. Integration by parts handles these by reversing the OTHER differentiation rule: the product rule.

The formula

Start from the product rule

\dfrac{d}{dx}(uv) = u\dfrac{dv}{dx} + v\dfrac{du}{dx}

and integrate both sides:

uv = \int u\dfrac{dv}{dx}\,dx + \int v\dfrac{du}{dx}\,dx

. Rearranged, that is the formula:

\int u\,\frac{dv}{dx}\,dx = uv - \int v\,\frac{du}{dx}\,dx

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