Oxford and Cambridge questions built from compound-angle identities, an inverse-trig derivative and a definite integral that has to be solved for algebraically once it loops back on itself — worked here in full, alongside the six-point method this pack uses to sketch a curve nobody has shown you before.
A pack this specific is unusual: seventeen pages, and every one of the ten questions belongs to just two skills, curve sketching or integration, rather than a scattershot of Oxbridge maths topics. The Integration & Curve Sketching pack pairs a Gaussian bell curve and a half-angle tangent substitution against the six-stage sketching checklist, working both from bare question through to full solution, so each habit sharpens alone before an interviewer asks you to combine them.
The Integration & Curve Sketching pack — £180
Ten questions across 17 pages. Each begins as the question alone, adds the hints for when your first attempt runs dry, then ends with a full worked answer. One PDF, one payment, instant download.
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A sketch and an integral test different things, and this pack keeps them apart rather than always forcing them together. Three of its ten questions are pure curve sketching, each one built from the same six checks run in the same order — where the curve meets the axes, where it is undefined, what happens as x runs to plus or minus infinity, where it turns, whether it is odd, even or neither, and where it inflects — applied to three unrelated functions so the method becomes a reflex rather than a fact to recall. The other seven are integration and its close neighbours: a trig identity carried through to an exact value, an inverse-trig derivative found by differentiating the wrong way round, and the pack’s one real trick used twice in different disguises — an integral that reappears on the far side of its own equation after a second pass of integration by parts and has to be solved for algebraically, and a definite integral pinned down the same way by a substitution that swaps its limits and sends the integral back to minus itself. The Integration & Curve Sketching pack, £180, is one of the shortest of the thirty — ten questions over seventeen pages — because it drills these two habits on their own, before either is asked to carry the other.
Because a demonstration stopped halfway proves nothing about either habit. A checklist sketch is only as convincing as the check you did not skip, and a self-referential integral is not solved until the loop actually closes and gets set equal to itself — stop one line early on either and you are left with a guess, not an answer. Both demonstrations below are taken to the end for that reason. None of it is a look inside the file: there is no free sample of this pack anywhere on the site and no form that will produce one, and the samples that do exist are for nine other subjects. It is bought unseen, and that is the reason for writing two full demonstrations rather than describing them.
The combination of integration and curve sketching in a single question is not arbitrary. At undergraduate level — and specifically in the first year of a Mathematics or Physics degree at Oxford or Cambridge — students routinely need to sketch a function to understand its behaviour before integrating it to find areas, volumes, or physical quantities. The interview tests whether you have already developed this habit of mind: the instinct to draw before you compute.
When asked to find the area enclosed between two curves, a student who sketches both curves first will immediately see which is on top, where they intersect (giving the limits of integration), and whether the enclosed region is above or below the x-axis. A student who goes directly to algebra risks integrating in the wrong order, using incorrect limits, or obtaining a negative area and not noticing the error. The sketch is not decoration; it is an essential step in the integration problem.
Conversely, interviewers sometimes present an integral and ask the candidate to interpret it geometrically — to draw the region whose area the integral represents, or to reason about the integral's sign and approximate magnitude from a sketch rather than computing it. This direction — from integral to sketch — tests whether you understand integration as a geometric concept rather than just a computational procedure. This particular pack keeps the two directions separate rather than combining them: three questions are sketches on their own terms, the rest are integration and algebra on their own terms, and the connecting habit is left for you to build once both are solid.
A-level integration covers substitution, by-parts, and standard forms. Oxbridge interviews go further — not in the sense that they require knowledge of university-level techniques, but in the sense that they test flexible and creative application of A-level techniques to unfamiliar integrals. The key integration methods tested are as follows.
Integration by parts is tested repeatedly, often in cascading form: ∫xⁿeˣdx requires integration by parts applied n times, and the student who understands why (because each application reduces the power of x by one) will be faster and more confident than one who has only practised single applications. Interviewers also test the case where integration by parts loops: ∫eˣsin(x)dx requires integration by parts twice to return to the original integral, which can then be solved algebraically as an equation. Recognising when this loop occurs — and knowing to solve for the original integral rather than continuing to apply parts — is a significant differentiator.
Trigonometric substitution appears in integrals involving √(a²−x²), √(x²+a²), and √(x²−a²). For √(a²−x²), the substitution x = a sin θ transforms the integral into one involving cos θ, which is typically straightforward. For √(x²+a²), use x = a tan θ; for √(x²−a²), use x = a sec θ. The student who knows which substitution to apply in each case, and can explain why (because the relevant trigonometric identity eliminates the square root), demonstrates real command of integration technique.
Partial fractions are required for integrands that are rational functions with factorisable denominators. For a denominator with two distinct linear factors, the decomposition is standard. For repeated factors, a repeated fraction term is needed. For irreducible quadratic factors, a linear numerator is required. Interviewers sometimes present integrals where the degree of the numerator is equal to or greater than the degree of the denominator — which requires polynomial long division before partial fractions can be applied. Getting this step right unprompted is a significant marker of mathematical sophistication.
| Integration Technique | When to Apply | Common Pitfall |
|---|---|---|
| Integration by parts | Products of polynomials with exp/trig/log | Forgetting to differentiate u or integrate dv |
| Substitution (algebraic) | Composite functions; u simplifies the integral | Forgetting to substitute dx or transform limits |
| Trig substitution | Integrals with √(a²±x²) or √(x²−a²) | Incorrect choice of substitution; forgetting to back-substitute |
| Partial fractions | Rational functions with factorisable denominators | Not doing long division first if degree(num) ≥ degree(denom) |
| Recognition of standard forms | ∫1/(x²+a²)dx = (1/a)arctan(x/a)+c | Confusing with ∫1/√(a²−x²)dx = arcsin(x/a)+c |
| Completing the square | Quadratic in denominator that won't factorise | Sign errors in completing the square step |
The stage-by-stage method set out on our graph sketching page is the same one this pack runs, question after question, until it stops needing to be looked up. Six checks, always in the same order: where the curve crosses the axes; where it is undefined, and whether that leaves an asymptote; what it does as x runs to plus and minus infinity; where it turns, found by differentiating; whether the function is odd, even or neither; and where it inflects. None of the six is optional — skip the symmetry check and you sketch twice the work that was needed, skip the behaviour-at-infinity check and you draw an asymptote where there is none.
This pack runs that same six-point method against three functions that have nothing else in common — one built from a negative exponential of a square, one from the logarithm of a quadratic, one a polynomial multiplied by an exponential — so what gets practised is the checklist itself, not a memorised shape. One of the three turns out, once drawn, to be the outline of a distribution most sixth-formers already recognise; the other two do not resemble anything from A-level at first glance, which is rather the point of being asked for them.
One question, run through all three sections in the order this pack uses them, and taken to the last line.
Show that ∫₀^(π/4) ln(1 + tan x) dx = (π/8) ln 2.
Call the integral I, and substitute x = π/4 − u. As x runs from 0 to π/4, u runs from π/4 down to 0; the minus sign from dx = −du cancels against reversing the limits back the right way round, so I = ∫₀^(π/4) ln(1 + tan(π/4 − u)) du — same limits, same value, a different-looking integrand.
Expand tan(π/4 − u) with the compound-angle formula: tan(π/4 − u) = (1 − tan u)/(1 + tan u). So 1 + tan(π/4 − u) = [(1 + tan u) + (1 − tan u)] / (1 + tan u) = 2/(1 + tan u), and the log of that splits as ln 2 − ln(1 + tan u).
Substitute that back in: I = ∫₀^(π/4) [ln 2 − ln(1 + tan u)] du = (π/4) ln 2 − ∫₀^(π/4) ln(1 + tan u) du. That last integral is I again. So I = (π/4) ln 2 − I, which gives 2I = (π/4) ln 2, and I = (π/8) ln 2.
Not a technique from the syllabus — there is no standard rule for integrating ln(1 + tan x) directly, and reaching for one wastes the first two minutes. What is being tested is recognising that a substitution mapping an interval back onto itself can turn an integral into an equation about itself, and having the discipline to stop trying to compute and start solving algebraically instead. It is the same move, in a different disguise, that this pack also uses inside a run of integration by parts: get the original expression to reappear on the far side of an equals sign, and treat it as an unknown rather than a target.
Three sections, kept apart on purpose.
The questions come first and bare, so one can be attempted with no hint anywhere in view. Then the hints: the follow-ups an interviewer would put to that particular question, in the order they would come. Then the suggested answers, written in the first person — someone reasoning, rather than a solution being displayed. The second and third sections are meant to be reached after your own attempt, and that is where the length of a pack goes: not into more questions, but into what is written underneath each one. Written by specialist subject tutors.
The Integration & Curve Sketching pack is £180 — one payment, downloadable and printable.
Get the Integration & Curve Sketching pack — £180Improper integrals — integrals with infinite limits or integrands with singularities within the interval of integration — appear in Oxford and Cambridge interviews as a test of whether candidates understand integration as a limit process rather than just a mechanical computation. The most important improper integrals are: ∫₁^∞ (1/xᵖ)dx (convergent for p > 1, divergent for p ≤ 1), ∫₀^1 (1/xᵖ)dx (convergent for p < 1, divergent for p ≥ 1), and ∫₋∞^∞ e^(−x²)dx (the Gaussian integral, equal to √π, proved using the polar coordinates trick).
The approach to any improper integral is the same: replace the problematic limit with a parameter (call it t or R), evaluate the definite integral with that parameter, and then take the limit as the parameter approaches the problematic value. If the limit exists and is finite, the integral converges; if not, it diverges. Stating this procedure explicitly — and narrating it as you apply it — demonstrates understanding of the foundational definition of integration as a limit rather than just a computational formula.
Interviewers sometimes ask for the physical interpretation of a convergent improper integral: what does it mean that ∫₁^∞ (1/x²)dx = 1? This is asking you to interpret the area under the curve y = 1/x² for x ≥ 1 as a finite number, which requires reconciling the intuition that an infinitely long strip must have infinite area with the mathematical fact that the decay of the function is fast enough to keep the total area bounded. Reasoning about this clearly — and connecting it to the comparison test for series — is the kind of discussion Oxford and Cambridge interview rooms are built for.
First-order differential equations with separable variables appear in Oxford and Cambridge Maths and Physics interviews as integration problems in disguise. A separable ODE has the form dy/dx = f(x)g(y), which can be rearranged as dy/g(y) = f(x)dx and integrated on both sides. The most common examples are: exponential growth/decay (dy/dx = ky, with solution y = Ae^(kx)), Newton's law of cooling (dT/dt = −k(T − T₀)), and simple harmonic motion (d²x/dt² = −ω²x).
What interviewers expect beyond simple separation of variables is the ability to apply initial conditions correctly, check the physical dimensions of the solution, and interpret the result. 'Does your solution make physical sense as t → ∞?' and 'what is the long-term behaviour of the temperature in Newton's law of cooling?' are standard follow-up questions. The curve sketching connection appears here too: once you have derived the solution, you should be able to sketch it and describe its qualitative behaviour — which is more informative for physical understanding than the formula alone.
Sketch y = x/(1 + x²) for all real x, using nothing but the checklist: intercepts, domain and asymptotes, behaviour as x → ±∞, turning points, symmetry, and inflection.
Intercepts: the only zero of the numerator is x = 0, and y(0) = 0, so the curve meets both axes once, at the origin. Domain: 1 + x² is never zero, so the function is defined for every real x and there is no vertical asymptote. Behaviour at infinity: dividing top and bottom by x² gives y = (1/x)/(1 + 1/x²), which → 0 as x → ±∞, so y = 0 is a horizontal asymptote in both directions. Turning points: by the quotient rule, dy/dx = (1 − x²)/(1 + x²)², zero at x = ±1, giving a maximum at (1, 1/2) and a minimum at (−1, −1/2). Symmetry: f(−x) = −f(x), so the function is odd — the curve has half-turn symmetry about the origin, which is why the maximum and minimum are mirror images through it. That is five of the six checks and already enough to draw it with confidence: rising from the horizontal asymptote at the far left, up through the origin to a peak just past x = 1, then back down through zero symmetry to a trough just past x = −1, and back out to the same asymptote on the right — skip the symmetry check and you would sketch the right-hand hump and then have to work out the left-hand one from scratch instead of reading it straight off.
"I had no idea what to expect from my Maths interview at Magdalen — A-level gives you no preparation for the style of question they ask. Working through the pack beforehand meant I'd practised thinking through problems I'd never seen before. When I got stuck in the actual interview, I knew how to keep going rather than freeze. I got my offer in January."— James H., Mathematics, Magdalen College Oxford, 2024 entry
“My panel at Gonville & Caius handed me a short article about a clinical trial and asked what I thought the key limitation was. I’d never seen the paper before. The pack was the only preparation I found that actually trains you for that — reading through the model answers showed me how to reason about evidence out loud, identifying what is missing or uncertain rather than just summarising what is there. By the time I got into the room I knew how to think, not just what to say.”— Priya S., Medicine, Gonville & Caius Cambridge, 2024 entry
"My tutor at Balliol pushed back on everything I said. The pack was the only resource I found that prepares you for that — the model answers show you how to structure an argument and defend it under pressure, not just state a view. Really glad I used it."— Ella T., History, Balliol College Oxford, 2025 entry
What £180 buys is length in one direction.
Not a longer list of questions — the list is short, ten questions in seventeen pages, one of the shortest of the thirty — but more written underneath each one: the hints an interviewer would actually give on that specific question, and a suggested answer written out in full rather than left as a final line. Three questions drill the six-point sketching checklist against unrelated functions; the rest are trig identities, an inverse-trig derivative and the pack's own loop trick, used twice, for pinning down an integral that would otherwise resist a direct attack. Written by specialist subject tutors.
Trustpilot puts Leading Tuition at 4.8 out of 5. Those reviews were left about tutoring and this is a document, so read them as evidence about who wrote it rather than about what you download. That is the company’s rating, not a rating of this pack.
Open the Integration & Curve Sketching pack — £180This particular pack keeps to a narrower set than that question usually implies: a compound-angle identity carried through to an exact surd value, an inverse-trigonometric derivative found by differentiating implicitly rather than quoting the result, a reduction of tan³x using tan²x + 1 = sec²x, and integration by parts pushed to a second pass so the original integral reappears and has to be solved for algebraically. There is no partial-fraction work in it and no classical trigonometric substitution for √(a²−x²)-type integrals — those belong to other packs in the range. What this pack tests is fluency with identities and derivatives rather than breadth of integration methods.
They are often paired so that the sketch fixes an area's limits and the integral gives the sketch numerical precision. This particular pack, however, keeps the two skills apart rather than combining them: three of its ten questions are sketches on their own terms, checked against a six-point method rather than an integral, and the rest are integration, trig identities and differentiation with no sketch involved at all. It suits a candidate building the two habits separately, before a harder pack asks for them together.
State your substitution explicitly, explain why it simplifies the integral, transform both the integrand and the differential completely before integrating, and for definite integrals, transform the limits as well. Example: for ∫x√(1−x²)dx, say 'I'll substitute u = 1−x², so du = −2x dx, meaning x dx = −du/2. The integral becomes −(1/2)∫√u du = −(1/3)u^(3/2) + c = −(1/3)(1−x²)^(3/2) + c.' This narration demonstrates understanding of what the substitution achieves, not just mechanical execution.
A frequently tested question type is: given the sketch of y = f(x), sketch y = f'(x) without differentiating algebraically. This requires knowing that f'(x) is zero at maxima and minima of f(x), positive where f(x) is increasing, negative where f(x) is decreasing, and has extrema at inflection points of f(x). The reciprocal — given f'(x), recover qualitative features of f(x) by graphical integration — also appears. Both questions test whether you understand calculus as a geometric operation rather than just an algebraic procedure.
Yes, generally — but not in this particular pack. Every one of its ten questions is pure mathematics: no forces, charges or physical quantities, just functions, identities and limits treated on their own terms. Integration set in a physical context belongs to our Physics and Engineering packs rather than this one.
Only partly, and it is worth being clear about which part. An integral has one value: if your answer is π/2 and the answer is π, you know at once, and you can go looking for the factor of two. A sketch has no single correct drawing. Two curves can look alike on paper and still disagree about something that matters — whether the shoulder sits inside or outside the turning point, whether two maxima are drawn level when the algebra makes one of them twice the height of the other — and holding your picture next to a printed one tells you the two pictures resemble each other, which is a different finding. So the Integration & Curve Sketching pack can mark your integration and it cannot mark your sketch. What it does instead is set the reasoning down in the order it was built, so the thing you compare is the route you took rather than the picture you ended up with.
Further Reading: Integration and curve sketching are core Maths interview skills. For the wider Oxford Maths interview format and additional worked examples, see our companion guide: Oxford Maths Interview Questions 2026 — Step-by-Step Model Answers.
See also: Maths Oxbridge Interview Questions, Graph Sketching Oxbridge Interview Questions, Derivations Oxbridge Interview Questions, and MAT preparation.
With a checklist you have not had to memorise yet and an integral that will not sit still under a direct attack. The two demonstrations on this page are the pack's shape in miniature: the same six checks run against an unfamiliar function, and the same self-referential trick used to pin down an integral that has no ordinary antiderivative. This pack drills both in isolation, ten questions over seventeen pages, by specialist subject tutors.
See the Integration & Curve Sketching pack — £180