Q: What does H2 Maths Notes (JC 1-2): 5.3) Integration Techniques cover? A: Substitution, parts, partial fractions, and special integrals for H2 Maths Topic 5.3.
Before you revise Build a flowchart for integration: recognise the pattern (substitution, parts, partial fractions, trigonometric identities) before diving in. Always write differential substitutions explicitly.
Core Formulae
Reverse of differentiation rules (power, exponential, trigonometric, logarithmic).
Integration by substitution: choose u=g(x) to simplify integrand, rewrite dx=g′(x)du
.
Integration by parts: ∫udv=uv−∫vdu.
Partial fractions: decompose rational functions into simpler fractions before integrating.
Substitution Examples
Example -- Trigonometric substitution
Evaluate ∫x2+12xdx.
Let u=x2+1⟹du=2xdx.
Integral becomes ∫u1du=ln∣u∣+C=ln∣x2+1∣+C.
Example -- Exponential substitution
Evaluate ∫xex2dx.
Let u=x2⟹du=2xdx.
Integral =21∫eudu=21ex2+C.
Integration by Parts
Choose u to simplify upon differentiation; set dv as remaining factor.
For repeated parts, loop until integral becomes elementary.
A handy heuristic for choosing u is the ILATE/LIATE mnemonic-use it to decide which factor to differentiate so the algebra keeps simplifying. It is widely taught even though the official H2 syllabus only specifies the integration-by-parts technique itself, so treat ILATE/LIATE as planning advice rather than examinable content.
Order
Pick this first when
I
an Inverse trigonometric term (e.g. arctanx) appears
L
a Logarithmic term (e.g. lnx) is present
A
an Algebraic factor (polynomial or rational) remains
T
Trigonometric or hyperbolic functions turn up
E
finally, the Exponential factor eg(x)
Example -- xcosx
Take u=xdv=cosxdx.
∫xcosxdx=xsinx−∫sinxdx=xsinx+cosx+C.
Example -- Reduction formula
For In=∫xnexdx, set u=xndv=exdx:
In=xnex−n∫xn−1exdx=xnex−nIn−1.
Partial Fractions
Proper rational functions: express as sum of linear/quadratic factors.
For repeated factors, include each power up to multiplicity.
For irreducible quadratics, use quadraticAx+B.
Example -- Partial fraction integral
Evaluate ∫x2−x−23x+5dx.
Factor denominator: (x−2)(x+1).
Decompose: (x−2)(x+1)3x+5=x−2A+x+1B.
Solve for A=2B=1.
Integrate: ∫(x−22+x+11)dx=2ln∣x−2∣+ln∣x+1∣+C
Trigonometric Identities
Use sin2x=21(1−cos2x), cos2x=21(1+cos2x), and sinxcosx=21sin2x.
For integrals of secx or tanx, multiply numerator and denominator to match derivatives.
Example -- ∫sec2x
Recognise derivative of tanx∫sec2xdx=tanx+C.
Improper Integrals and Convergence
Evaluate limits for infinite bounds or singularities:
∫1∞xp1dx converges iff p>1.
For improper points inside interval, split integral and take limits.
Calculator Workflow
Use a graphing calculator (GC) to confirm antiderivatives by differentiating results.
Numerical integration (∫f(x)dx) checks definite integrals; still show analytical steps.
Store substitution results before back-substituting to reduce mistakes.
Exam Watch Points
Explicitly state substitution and new limits for definite integrals.
Include constant of integration for indefinite integrals.
Handle absolute values in logarithms when integrating rational functions.
Justify convergence when dealing with improper integrals.
Practice Quiz
Reinforce substitution, parts, and partial-fractions strategies-including improper integral convergence checks.
Quick Revision Checklist
Identify the right technique (substitution, parts, partial fractions) for each integrand.
Carry out algebraic manipulations cleanly before integrating.
Evaluate definite integrals with correct limits and substitution steps.
Discuss convergence for improper integrals and document limit processes.
