MAST10006 · Calculus 2
Integration Techniques
Differentiation is mechanical; integration is pattern recognition. There is no single algorithm — you read the shape of the integrand and reach for the matching tool, and this chapter is the decision tree. The six moves are integration by parts (chosen by the LIATE priority order), partial fractions (which split a rational function into pieces that integrate to logs and arctans), trigonometric substitution and hyperbolic substitution (each of which uses a Pythagorean-style identity to kill a square root), the peel-and-identity method for powers of sin/cos and sinh/cosh, and improper integrals (where an infinite limit or an interior blow-up is replaced by a limit). Learn the trigger for each method and most exam integrals fall in one or two lines. The recurring marked points are choosing u correctly, solving algebraically for a return-of-integral, and writing the limit explicitly for any improper integral.
What this chapter covers
- 017.2 Integration by parts and the LIATE priority order
- 02Worked by-parts: power×exp, the lone log, return-of-integral
- 037.6 Partial fractions → logs and arctans
- 047.3 Trigonometric substitution
- 057.4-7.5 Hyperbolic substitution and powers of sin/cos/sinh/cosh
- 06Improper integrals — replace the bad point with a limit
Worked example: integration by parts (return-of-integral)
- +1First pass: u = ex, dv = cos x dx gives I = ex sin x − ∫ ex sin x dx.
- +1Second pass (same u-type): ∫ ex sin x dx = −ex cos x + ∫ ex cos x dx, and the original integral I reappears.
- +1Substitute back: I = ex sin x − (−ex cos x + I) = ex(sin x + cos x) − I.
- +1Solve algebraically: 2I = ex(sin x + cos x), so I = ½ ex(sin x + cos x) + C.
Key terms
- Integration by parts
- The product rule run backwards: ∫ u dv = uv − ∫ v du. Useful only when the new integral is simpler. The factor that gets differentiated (u) is chosen by LIATE so that it simplifies; the rest becomes dv.
- LIATE
- The priority order — Logarithm, Inverse-trig, Algebraic, Trig, Exponential — for choosing u in integration by parts. The factor earliest in the list becomes u (a log or inverse-trig differentiates away to something algebraic); an exponential or trig, which integrates back to itself, is happiest as dv.
- Partial fractions
- Rewriting a rational function P(x)/Q(x) as a sum of simpler terms, each integrating to a log or an arctan. Do long division first if the top degree is ≥ the bottom; a repeated factor needs a term for each power; an irreducible quadratic needs a full linear numerator Bx + C.
- Trigonometric / hyperbolic substitution
- Replacing x by a sinθ, a tanθ, a secθ (or a sinh u, a cosh u) so that a Pythagorean-style identity removes a square root of a quadratic. The choice is dictated by the root: √(a²−x²) → sine, √(a²+x²) → tangent or sinh, √(x²−a²) → secant or cosh.
- Improper integral
- An integral with an infinite limit of integration or an integrand that blows up inside the interval. It is evaluated by replacing the bad point with a variable and taking a limit; the integral converges if that limit is finite and diverges otherwise.
Integration Techniques FAQ
How do I decide which integration technique to use?
Read the integrand first. A bare log or arctan with no other factor means parts with dv = dx. A polynomial times an exponential, sine or cosine means parts (or the tabular shortcut). A product of two cycling functions like ex sin x means parts twice and solve for I. A rational function means partial fractions. A square root of a quadratic means a trig or hyperbolic substitution. A single composite f(g)·g′ is a plain substitution, not parts.
How do I pick u and dv in integration by parts?
Use LIATE: choose u as the factor that comes first in the list Logarithm → Inverse-trig → Algebraic → Trig → Exponential, because that factor differentiates toward something simpler. The remaining factor is dv. A lone log or arctan is integrated by taking the hidden dv = dx.
What is the return-of-integral trick?
For integrands like ex sin x, two applications of parts reproduce a multiple of the original integral I. Instead of looping forever, write the resulting equation I = … ± I and solve algebraically — 2I = … gives I = ½(…). Keep the same u-type both passes, and add +C only at the end.
What do I have to write for an improper integral?
Always write the limit explicitly — replace the infinite limit (or the blow-up point) by a variable, integrate, then take the limit. Plugging ∞ straight in loses marks even when the number is right. A blow-up inside the interval must be split at the bad point into two improper integrals, and the whole thing converges only if both halves do. State whether it converges or diverges; that is the question being marked.
Exam move
Drill the decision tree until the trigger-to-tool mapping is reflexive: product where one factor simplifies → parts (LIATE); rational function → partial fractions; root of a quadratic → trig or hyperbolic sub; powers of sin/cos or sinh/cosh → peel an odd factor or use a double-angle on even powers; infinite limit or interior blow-up → improper integral with an explicit limit. The single most valuable habit is to read the integrand and name the method before writing anything. For by parts, practise the return-of-integral solve and the lone-log dv = dx move; for partial fractions, never forget long division when top-heavy, a term per power for repeated factors, and a linear numerator for irreducible quadratics; for substitutions, build the right triangle (or recall the master identity) to back-substitute. For improper integrals, the limit line is non-negotiable and the converge/diverge verdict is what scores.
