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Integration by parts reverses the product rule: ∫u dv = uv − ∫v du. Choosing u to be the factor that simplifies on differentiation is what makes the remaining integral easier.
Integration by parts
∫u dv = uv − ∫v du, or [uv]ₐᵇ − ∫ₐᵇ v du for a definite integral
The integral is 4 − 4 = 0, showing how the boundary term and remaining integral can cancel.
The LIATE order — logarithmic, inverse trig, algebraic, trigonometric, exponential — usually picks a u that simplifies.