A Level · Cambridge (CIE) · 9709

Integrate natural exponential composites by reversing their inner scale

Divide by a non-zero linear inner coefficient, keep the integration constant and check accumulated quantities against their rate.

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 natural exponential composites with the chain rule.
Use an integration constant and evaluate a definite integral.

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

Quick readiness check

What is d(e^(3x))/dx?

What you will learn

Integrate constant multiples, sums and differences of e^(ax + b), including negative a.
Use an initial condition or definite integral and distinguish an accumulated increase from a total.

Reverse the exponential's chain multiplier

For a ≠ 0, ∫e^(ax + b) dx = e^(ax + b)/a + C. Differentiation multiplies by a, so integration divides by it. The exponential retains its complete original argument; it does not become a power with a new exponent. A negative inner coefficient produces a negative divisor. For example, ∫e^(−2x) dx = −e^(−2x)/2 + C, whose derivative is the positive integrand.

Integrate constant multiples and sums term by term. Thus ∫[3e^(2x) − 4e^(−x)] dx = (3/2)e^(2x) + 4e^(−x) + C. The second sign becomes positive because the outside −4 is divided by inner coefficient −1. One arbitrary constant suffices for the whole antiderivative on its connected domain. If a = 0, the integrand eᵇ is constant and its integral is eᵇx + C; do not divide by zero.

Check both a derivative and any supplied condition

If y′ = 6e^(2x) and y(0) = 5, first obtain y = 3e^(2x) + C. The initial value gives 5 = 3 + C, so y = 3e^(2x) + 2. Differentiation checks the rate, while substitution checks the original value; these are separate requirements. A definite integral uses F(b) − F(a), so the same additive constant cancels.

For a supplied inflow q(t) = 4e^(t/2) litres per minute on 0 ≤ t ≤ 2, an antiderivative is 8e^(t/2). The added volume is 8(e − 1) litres. If a vessel starts with 7 litres and has no outflow, its final total is 7 + 8(e − 1) = 8e − 1 litres. Integrating a rate gives an increase; an initial stock must be added separately when the question asks for a total.

Pause and explain

How should ∫e^(0x + b) dx be handled?

Put the idea to work

Worked example

Integrate 3e^(2x) − 4e^(−x) and verify the signs by differentiation.

Show the worked solution

Divide the first coefficient by inner rate 2 to obtain (3/2)e^(2x).
Divide the second coefficient −4 by inner rate −1 to obtain +4e^(−x).
Add C and differentiate both terms, recovering 3e^(2x) − 4e^(−x).

Answer (3/2)e^(2x) + 4e^(−x) + C.

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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

Integrate 5e^(3x + 1) − 2e^(−4x) and check your answer.

Give me a hint

Divide each coefficient by its own signed inner rate.

Compare my reasoning

The first term integrates to (5/3)e^(3x + 1).
The second term integrates to (1/2)e^(−4x), because −2/(−4) = 1/2.
Add C; differentiation gives 5e^(3x + 1) − 2e^(−4x), recovering both signs.

(5/3)e^(3x + 1) + (1/2)e^(−4x) + C.

Look for these in your work

Retain each original exponential argument and signed inner scale.
Verify both terms by differentiation and retain C.

2 · Independent

A curve has y′ = 6e^(2x) and passes through (0, 5). Find its equation and check both conditions.

Give me a hint

Find an antiderivative before substituting the original point.

Compare my reasoning

Integration gives y = 3e^(2x) + C.
At x = 0, 5 = 3 + C, so C = 2.
The curve y = 3e^(2x) + 2 has the required derivative and value 5 at zero.

y = 3e^(2x) + 2.

Look for these in your work

Reverse the inner factor 2 before determining C.
Check both the derivative and the original point.

3 · Transfer

A vessel initially holds 7 litres. Its supplied inflow is q(t) = 4e^(t/2) litres per minute for 0 ≤ t ≤ 2, with no outflow. Find the added volume and final total exactly.

Give me a hint

Use a definite integral for the increase, then include the initial stock only in the total.

Compare my reasoning

An antiderivative of q is 8e^(t/2).
The added volume is 8e − 8 = 8(e − 1) litres.
Adding the initial 7 litres gives final total 8e − 1 litres.

Added volume 8(e − 1) litres; final total 8e − 1 litres.

Look for these in your work

Use the interval endpoints in the correct order.
Distinguish accumulated increase, initial stock and final total with matching units.

Common mistakes

Reverse the inner multiplier by division, including its sign.
Keep C for an indefinite integral, and use a supplied value only after integrating.
An integral of a flow rate gives added volume, not automatically the final stock.

Recall without your notes

What can you explain now?

Find ∫e^(−2x) dx.

Compare with the explanation

−e^(−2x)/2 + C.

Differentiating multiplies by −2, cancelling the divisor and returning the original positive exponential.

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Make it stick

Explain why a linear exponential integral divides by its inner coefficient. Check a negative-coefficient example and distinguish a flow integral from a final stored quantity.

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.