Integration Techniques — Substitution, Parts, Partial Fractions

GRE Math Subject · Axiom Academy

Master the three most common integration methods Method: Let u = g(x), du = g'(x) dx When to use: The integrand contains a composite function where the inner function's derivative (or a constant multiple) appears. Let u = x² + 1, du = 2x dx ⟹ = Mnemonic (LIATE): Choose u in this order of priority: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential u = x (algebraic), dv = e^x dx Advanced: You may need integration by parts twice. Keep your work organized. Partial Fractions Decomposition When to use: Rational functions (polynomial/polynomial). Decompose into simpler fractions. If degree of numerator ≥ degree of denominator, do polynomial long division first Set up partial fraction form with unknowns A, B, C, ... Multiply both sides by denominator and solve for constants Integrate each simple fraction Setting x=1: 2 = 2A ⟹ A=1; Setting x=-1: 2 = -2B ⟹ B=-1

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