A Level · Cambridge (CIE) · 9709

Integrate linear partial fractions including repeated factors

Decompose first, integrate simple and repeated poles with different rules, and preserve the original excluded inputs.

Section 3.5 · Integration

Pure Mathematics 3 · Integration · 3.5

Exam scope and assessment notes

Pure Mathematics 3 (Paper 3, A Level) · Mathematics 9709, 2026–2027, section 3.5. Paper 1 integration, shared standard integrals and identities, Paper 3 partial fractions and derivatives are assumed. Rational decompositions follow the three denominator forms in section 3.1; higher-degree numerators are excluded, while equal-degree cases may need division first. Substitution questions use a given substitution. Trigonometric and inverse-tangent outputs use radians. Written practice is self-checked; human subject-teacher review remains pending.

Before you begin

Decompose the linear-factor forms in Paper 3 Algebra.
Integrate linear reciprocals with logarithms and other negative powers with the power rule.

Work on paper. Try each question before revealing help, and explain why your method works.

Quick readiness check

Which terms are needed for a denominator x(x + 1)²?

What you will learn

Integrate rational functions with distinct or repeated linear factors after a valid decomposition.
Handle an equal-degree numerator by division and check the full definite interval for poles.

The denominator power decides the primitive

Resolve a rational function into the allowed partial fractions before integrating its terms. For a ≠ 0, ∫A/(ax + b) dx = (A/a)ln|ax + b| + C on each connected interval avoiding its pole. For a repeated square, ∫B/(ax + b)² dx = −B/[a(ax + b)] + C. The square needs a reciprocal primitive rather than the same logarithm used for the first power.

For (3x + 5)/[(x + 1)(x + 2)], the decomposition is 2/(x + 1) + 1/(x + 2). Recombining gives numerator 2(x + 2) + (x + 1) = 3x + 5, an independent algebra check. Integration gives 2ln|x + 1| + ln|x + 2| + C on an interval excluding −1 and −2. The absolute values provide positive log arguments; they do not remove the original poles.

Division and endpoint checks are separate steps

An equal-degree case such as x²/(x² − 1) first becomes 1 + 1/(x² − 1), then 1 + 1/[2(x − 1)] − 1/[2(x + 1)]. Its primitive is x + (1/2)ln|(x − 1)/(x + 1)| + C, for x ≠ ±1 on separate intervals. The polynomial term x would be lost if you treated the original function as a proper fraction without dividing. Section 3.1 excludes higher-degree numerators; this example uses equal degrees.

Before evaluating F(b) − F(a), check that the original denominator never vanishes between the endpoints. For ∫ from 2 to 3 of x²/(x² − 1) dx, there is no pole inside the interval, so the result is 1 + (1/2)ln(3/2). For an interval crossing 1 or −1, ordinary endpoint subtraction is not justified. When a factor cancels algebraically, a missing original input remains excluded unless the problem explicitly defines a continuous extension.

Pause and explain

What should happen before integrating x²/(x² − 1)?

Put the idea to work

Worked example

Integrate (3x + 5)/[(x + 1)(x + 2)] and state its exclusions.

Show the worked solution

Write A/(x + 1) + B/(x + 2), giving A + B = 3 and 2A + B = 5.
Hence A = 2 and B = 1; check by recombining the two fractions.
Integrate the two first-power reciprocals and retain the poles x = −1 and −2.

Answer 2ln|x + 1| + ln|x + 2| + C, on separate intervals excluding −1 and −2.

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From guided practice to a new situation

Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.

1 · Guided

Evaluate ∫ from 1 to 2 of 1/[x(x + 1)(x + 2)] dx exactly.

Give me a hint

Clear the three distinct linear factors to find coefficients 1/2, −1 and 1/2; check every pole is outside the interval.

Compare my reasoning

Recombination verifies 1/[x(x + 1)(x + 2)] = 1/(2x) − 1/(x + 1) + 1/[2(x + 2)].
A primitive on this positive interval is (1/2)ln x − ln(x + 1) + (1/2)ln(x + 2).
Subtract endpoints to get (5/2)ln 2 − (3/2)ln 3 = (1/2)ln(32/27); the poles 0, −1 and −2 are outside [1, 2].

(1/2)ln(32/27).

Look for these in your work

Verify all three fraction coefficients and the pole-free interval.
Subtract complete endpoints in the right order and simplify the logarithm.

2 · Independent

Integrate (x + 2)/[x(x + 1)²], giving all repeated-factor terms and exclusions.

Give me a hint

Clear the denominator and compare coefficients, or substitute the roots and one additional input.

Compare my reasoning

The decomposition is 2/x − 2/(x + 1) − 1/(x + 1)²; recombination gives x + 2.
Integrate term by term: 2ln|x| − 2ln|x + 1| + 1/(x + 1) + C.
Differentiation recovers the three fractions; retain original exclusions x = 0 and −1 on separate intervals.

2ln|x| − 2ln|x + 1| + 1/(x + 1) + C, with x ≠ 0, −1.

Look for these in your work

Include both powers of the repeated factor.
Use the correct reciprocal sign and preserve the original domain.

3 · Transfer

A supplied positive rate is q(t) = t²/(t² − 1) litres per minute for 2 ≤ t ≤ 3 minutes. Find the modelled volume added over this interval and explain why endpoint subtraction is valid.

Give me a hint

Divide the equal-degree fraction before decomposing its remainder.

Compare my reasoning

Write q(t) = 1 + 1/[2(t − 1)] − 1/[2(t + 1)].
A primitive is t + (1/2)ln|(t − 1)/(t + 1)|; neither pole ±1 lies in [2, 3].
Evaluation gives 1 + (1/2)ln[(2/4)/(1/3)] = 1 + (1/2)ln(3/2) litres.

Added volume 1 + (1/2)ln(3/2) litres.

Look for these in your work

Retain the divided polynomial part and all fraction coefficients.
Check poles and interpret a rate integral with matching time units.

Common mistakes

A repeated square integrates by the power rule, not by the same logarithm as a simple pole.
Keep the polynomial term after equal-degree division.
A finite pair of endpoint primitives cannot justify crossing an interior pole.

Recall without your notes

What can you explain now?

Find ∫1/(x + 1)² dx.

Compare with the explanation

−1/(x + 1) + C, with x ≠ −1.

Differentiating the negative reciprocal gives the positive inverse square.

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Make it stick

Explain the different primitives for simple and repeated linear factors. Decompose, integrate, differentiate back and check the interval before evaluating endpoints.

If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.

Keep learning Mathematics

Check the official syllabus 2026–2027

Course scope and assessment

312 sequenced lessons provide 936 guided, independent and application problems covering the 160 objectives in sections 1.1–1.8, 2.1–2.6, 3.1–3.9, 4.1–4.5, 5.1–5.5 and 6.1–6.5. The map lists all 38 sections across six alternative components; all eight Pure Mathematics 1 sections and all six Pure Mathematics 2 sections, from Paper 2 Algebra through Numerical solution of equations, currently have full teaching sequences. Paper 3 Algebra, section 3.1, links the six shared foundation lessons plus six lessons on partial fractions and rational binomial expansions; Paper 3 logarithms, trigonometry and numerical equations use the shared lessons, with the Paper 1 graph foundation linked into trigonometry. Paper 3 differentiation combines the shared methods with its inverse-tangent derivative lesson. Paper 3 Integration links four shared standard and identity lessons plus seven advanced lessons, covering all six section 3.5 requirements. Eight Vectors lessons cover all six section 3.7 requirements, with 2D and 3D line and solid applications. Nine Differential equations lessons cover all four section 3.8 requirements, including Paper 3 integration techniques, initial conditions, equilibrium solutions and conditional model interpretations. Eleven Complex numbers lessons cover all eight section 3.9 requirements, with seven Argand diagrams and explicit argument and boundary checks. All nine Pure Mathematics 3 sections now have teaching linked to every mapped requirement. Nine Forces and equilibrium lessons cover all seven section 4.1 requirements, with six force diagrams, signed incline-friction intervals, reaction-dependent pull and push limits and separate-body third-law checks. Nine Kinematics lessons cover the four official section 4.2 requirements, with the graph requirement mapped to three relationships. Six motion diagrams support directed quantities, graph slopes and signed areas, Paper 1 time-calculus methods, reversal distances, constant-acceleration stages and shared-clock particle meetings. Six Momentum lessons cover both section 4.3 requirements, including signed system totals, conservation assumptions, outgoing velocities, coalescence, unknown masses and successive impacts. Four diagrams show system momentum through the stipulated events, and supplied kinetic-energy checks distinguish conservation of momentum from changes in kinetic energy. Twelve Newton’s laws lessons cover all four section 4.4 requirements, with vertical support and release stages, rough horizontal and incline motion, inferred parameters, hanging and surface pulley systems, rigid tow-bar thrust and slack-string transitions. Seven force diagrams support the separate-particle balances. Nine Energy, work and power lessons cover all five section 4.5 requirements, with angle-dependent work, kinetic and reference potential energy, signed net-work balances, smooth curved slides, rough and driven slopes, ideal connected-system energy and instantaneous power on hills. Five diagrams distinguish the energy inputs, losses and valid positive-speed force model. All five Mechanics sections now have teaching and staged practice linked to their mapped requirements. Thirteen Representation of data lessons cover all five section 5.1 requirements, including choice and limitations of displays, ordered and back-to-back stems, raw summaries and box plots, unequal-width density histograms, cumulative-frequency interpolation, descriptive moment calculations, grouped estimates, coding, combining two datasets and correcting totals. Six accessible diagrams retain repeated observations and distinguish numerical spread, frequency area and estimated readings. Ten Permutations and combinations lessons cover both official section 5.2 requirements, with selection and ordered roles, repeated objects, adjacency blocks, non-adjacency gaps, leading-zero cases, labelled multi-row seating and mixed restrictions. Six accessible diagrams show outcome trees, ordered versus unordered choices, repeated copies, blocks, gaps and distinct row seats. Eleven Probability lessons cover all four section 5.3 requirements, with equiprobable elementary outcomes, selection-based probabilities, complements and disjoint regions, independence tests, restricted sample spaces, two-way tables, replacement and removal trees, reversed conditions and unknown parameters. Seven accessible diagrams show dice outcomes, random subsets, event regions, a conditioned grid, a count table and two probability trees. Twelve Discrete random variables lessons cover all three official section 5.4 requirements, with complete probability tables, moment and parameter checks, binomial assumptions, exact probabilities and moments, geometric first-success probabilities, tails, expectation and integer reliability thresholds. Six accessible diagrams distinguish outcome grouping, weighted moments, fixed trials, discrete probability bars, waiting conventions and integer tail deadlines. Twelve Normal distribution lessons cover all three official section 5.5 requirements, with continuous-area interpretation, MF19 cumulative tables, full standardisation, tail and interval regions, conditional probabilities, percentiles, one- and two-parameter inference, binomial suitability, continuity corrections and approximate integer thresholds. Seven accessible diagrams show model areas, table reading, aligned scales, intervals, quantiles, matched binomial and normal shapes and whole integer cells. All five Statistics 1 sections now have teaching and staged practice linked to their mapped requirements. Fourteen Poisson distribution lessons cover all five official section 6.1 requirements, with homogeneous event-count assumptions, exposure scaling, exact masses and tails, conditional count events, moments, parameter inference, probability duration targets, disjoint and overlapping windows, rare-count binomial approximations, matched normal approximations and continuity corrections. Nine accessible diagrams show event exposures, unit conversions, discrete bars, complete tail complements, population moments, shared windows, approximate count models and integer cells. Exact checks distinguish approximate and exact close-target decisions. Fourteen Linear combinations lessons cover the six listed results in official section 6.2, with affine moments, signed scales and constant cases, expectation without independence, independent squared variance contributions, fixed and random fees, independent copies versus reused values, affine normal events, independent normal sums and differences, weighted and conditional areas, coefficient inference, exact independent Poisson sums and overlapping exposure limits. Nine accessible diagrams distinguish scale changes, variance contributions, component costs, copy mechanisms, aligned normal scales, summed areas, Poisson count pairs and unsupported distribution claims. Fourteen Continuous random variables lessons cover both official section 6.3 requirements, with density validity and units, support-aware intervals, complements and conditional regions, first and second moments, percentiles by direct density areas, symmetry checks, parameter inference, infinite-support power and exponential integrals, divergent-moment limits and combined applications. Nine accessible diagrams distinguish density height from probability, normalisation, negative-density failures, central and conditional areas, mean versus median, percentile locations and partial infinite tails. Explicit cumulative distribution function knowledge is not required. Eighteen Sampling and estimation lessons cover all eight official section 6.4 requirements, with population and frame definitions, random-number selection, sampling-quality limits, independent sample-mean moments, normal means, CLT conditions, SE scaling, raw, summarised and coded unbiased estimates, known-variance and large-sample mean intervals, repeated-sampling interpretation, precision planning, proportion intervals, sparse-outcome limits and contextual applications. Ten accessible diagrams distinguish selection frames, number rejection, repeated sample means, population versus mean density, CLT models, variance divisors, interval targets, illustrative coverage, integer size checks and proportion margins. Eighteen Hypothesis tests lessons cover all five official section 6.5 requirements, with pre-specified alternatives, inclusive count tails, explicit equal-tail conventions, integer critical regions and actual significance, exposure-aware Poisson tests, continuity-corrected count approximations, normal and large-sample mean tests, confidence-interval connections, error classifications, fixed-region binomial, Poisson and normal error probabilities and sample-size power comparisons. Nine accessible diagrams show the testing procedure, exact count regions, unequal Poisson tails, discrete significance jumps, corrected count boundaries, matching mean intervals and alternative error distributions. All shared lessons are counted once in the course total. All 38 sections now link teaching and practice. Eight earlier overview lessons remain available, but do not count as full section coverage. This is a content-availability measure, not full programme completion or exam readiness. Written practice is self-checked, not officially marked. Check the 2028–2030 syllabus separately if sitting an examination from 2028; this teaching map uses 2026–2027 requirements.