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How to Use Integration by Parts
Integration by parts is the integral counterpart of the product rule for derivatives. It is the go-to technique for integrating products of unlike functions: polynomial × exponential, polynomial × trig, and anything involving ln(x). Master LIATE for picking u and the technique becomes mechanical.
When the method applies
- • Integrand is a product of two unrelated functions
- • u-substitution does not simplify anything
- • You see x·eˣ, x·sin(x), x·ln(x), or x²·eˣ patterns
- • Tabular method might apply
Common mistakes
- • Wrong choice of u and dv
- • Did not apply LIATE rule for picking u
- • Forgot the minus sign on ∫v du
- • Need to repeat the technique 2-3 times
Step-by-step method
- • Formula: ∫u dv = uv - ∫v du
- • Pick u using LIATE: Logarithms, Inverse trig, Algebraic, Trig, Exponential
- • For polynomials × exponentials/trig, use tabular integration (much faster)
- • If you get the original integral back, solve algebraically (∫ = expression / 2)
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Build long-term fluency
- • Drill LIATE on 20 problems until u-choice is automatic
- • Practice tabular integration for polynomial × exponential cases
- • Recognise the cyclic pattern (eˣ·sin(x), eˣ·cos(x))
Edge cases & deeper reading
If integration by parts spirals into more terms each step, you picked u and dv backwards — restart with LIATE. For definite integrals, do not forget to evaluate uv at the bounds.
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