A Level · Cambridge (CIE) · 9709

Integrate linear reciprocals with logarithms on valid intervals

Use an absolute value to describe either side of a reciprocal's pole, keep the inner divisor and never integrate straight across a singularity.

Section 2.5 · Integration

Pure Mathematics 2 & 3 · Standard integration and identities · 2.5 / 3.5

Exam scope and assessment notes

Pure Mathematics 2 (Paper 2, AS only) and Pure Mathematics 3 (Paper 3, A Level) · Mathematics 9709, 2026–2027. Shared sections 2.5 and 3.5 use these standard exponential, reciprocal and trigonometric integrals and identity methods. Paper 1 integration and the shared derivatives and identities are assumed. The AS route using Papers 1 + 2 cannot be carried forward to A Level. The general method of integration by substitution is not required for Paper 2; Paper 3 uses given substitutions in its separate lesson. Paper 3 also adds positive-quadratic reciprocals, partial fractions, logarithmic derivative patterns and integration by parts. Use radians for trigonometric calculus. Written practice is self-checked; human subject-teacher review remains pending.

Before you begin

Differentiate logarithms and identify positive log arguments.
Find reciprocal exclusions and evaluate a definite integral.

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

Quick readiness check

Why can the ordinary power integration formula not integrate x^(−1)?

What you will learn

Find an antiderivative of 1/(ax + b) for a ≠ 0 on a valid connected interval.
Evaluate definite logarithmic integrals and recognise when a pole prevents ordinary endpoint evaluation.

A reciprocal needs a logarithmic antiderivative

The power rule ∫xⁿ dx = x^(n + 1)/(n + 1) fails at n = −1. Instead, ∫1/x dx = ln|x| + C for x ≠ 0, applied on either connected interval x < 0 or x > 0. On the negative interval, ln|x| = ln(−x) has derivative (−1)/(−x) = 1/x. The absolute value supplies a positive log argument without making the original reciprocal defined at zero.

For a ≠ 0, ∫1/(ax + b) dx = ln|ax + b|/a + C where ax + b ≠ 0. Thus ∫6/(3x − 2) dx = 2 ln|3x − 2| + C. When the problem already restricts 3x − 2 > 0, ln(3x − 2) is an equivalent form on that interval. The constants on separate sides of the pole can differ; one initial value on one side does not determine the other side.

Check the entire integration interval before subtracting endpoints

For ∫ from −2 to −1 of 3/(2x − 1) dx, the denominator is negative but never zero. The antiderivative is (3/2)ln|2x − 1|, so the signed result is (3/2)[ln 3 − ln 5] = (3/2)ln(3/5), which is negative as expected from a negative integrand. The absolute value does not turn a signed integral into positive geometric area.

For ∫ from 0 to 1 of 1/(2x − 1) dx, the pole x = 1/2 lies inside the interval. Ordinary endpoint subtraction would misleadingly give zero because both endpoint logarithms are ln 1. Each one-sided improper integral diverges, so this ordinary improper integral does not exist. Cancellation of two infinities is invalid. A valid primitive on separated intervals cannot justify crossing their excluded input. If a = 0 and b ≠ 0, integrate the constant 1/b directly; if b = 0 too, the integrand is undefined.

Pause and explain

Does endpoint subtraction prove ∫ from 0 to 1 of 1/(2x − 1) dx equals zero?

Put the idea to work

Worked example

Find ∫6/(3x − 2) dx and state its interval restrictions.

Show the worked solution

The original reciprocal excludes x = 2/3.
Reverse the log chain factor: 6/3 = 2, giving 2 ln|3x − 2| + C.
Differentiation gives 2·3/(3x − 2), confirming the integrand on either separate domain interval.

Answer 2 ln|3x − 2| + C, on x < 2/3 or x > 2/3 separately.

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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

Find ∫4/(2x + 1) dx and state the excluded input.

Give me a hint

Use a logarithm, divide by the inner coefficient and check the reciprocal domain.

Compare my reasoning

The pole is at x = −1/2.
The antiderivative is 2 ln|2x + 1| + C on either separate interval.
Differentiation gives 2·2/(2x + 1) = 4/(2x + 1).

2 ln|2x + 1| + C, with x ≠ −1/2 and separate interval constants.

Look for these in your work

Reverse the inner log derivative by dividing by 2.
Preserve the reciprocal exclusion after integration.

2 · Independent

Evaluate ∫ from −2 to −1 of 3/(2x − 1) dx exactly and explain its sign.

Give me a hint

The interval stays entirely on one side of the pole; use the absolute values of both negative denominators.

Compare my reasoning

A primitive is (3/2)ln|2x − 1|, and no pole occurs on [−2, −1].
Endpoint subtraction gives (3/2)(ln 3 − ln 5) = (3/2)ln(3/5).
The result is negative, agreeing with the negative integrand throughout the interval.

(3/2)ln(3/5), a negative signed integral.

Look for these in your work

Verify continuity on the full interval before endpoint evaluation.
Retain signed accumulation despite the logarithmic absolute value.

3 · Transfer

A vessel starts with 5 litres and has no outflow. Its supplied inflow is 6/(t + 2) litres per minute on 0 ≤ t ≤ 4. Find the increase and final total exactly.

Give me a hint

The denominator stays positive on the time interval; integrate the rate and then add the initial stock.

Compare my reasoning

A primitive of the rate is 6 ln(t + 2), valid throughout the interval.
The increase is 6(ln 6 − ln 2) = 6 ln 3 litres.
The final total is 5 + 6 ln 3 litres; the reciprocal pole at t = −2 is outside the supplied model domain.

Increase 6 ln 3 litres; final total 5 + 6 ln 3 litres.

Look for these in your work

Use a valid connected interval and a correct logarithmic difference.
Report the increase and total as separate volumes.

Common mistakes

Do not apply the power formula with new exponent zero.
An absolute value in the log keeps its argument positive; it does not remove the original pole.
Check for a singularity inside a definite integral before evaluating endpoints.

Recall without your notes

What can you explain now?

Find ∫1/(3 − 2x) dx on a valid interval.

Compare with the explanation

−(1/2)ln|3 − 2x| + C, with x ≠ 3/2.

The logarithmic chain factor −2 is reversed by the divisor −2.

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Make it stick

Derive the reciprocal-linear log formula by differentiating it. Explain the absolute value and why a pole between the endpoints invalidates ordinary endpoint subtraction.

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.