A Level · Cambridge (CIE) · 9709

Use trigonometric identities to expose integrable terms

Rewrite squared sines, squared cosines, products and tangent squares before applying the standard primitives.

Section 3.5 · 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

Use double-angle identities and tan² u = sec² u − 1.
Integrate linear trigonometric composites in radians.

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

Quick readiness check

Which identity rewrites cos² u?

What you will learn

Use double-angle and reciprocal identities to rewrite a trigonometric integrand.
Check the transformed primitive and evaluate exact positive or signed integrals.

The blue curve is y = cos² 2x on 0 ≤ x ≤ π/4, with radian x. The shaded region is its positive area above the x axis. Its height is 1 at zero, 1/2 at π/8 and 0 at π/4. The identity cos² 2x = (1 + cos 4x)/2 gives primitive x/2 + sin 4x/8, so the exact shaded area is π/8. The curve continues outside this window; its shape is not an approximation by trapezia.

Change the integrand before trying to reverse differentiation

The identity sin² u = (1 − cos 2u)/2 and its companion cos² u = (1 + cos 2u)/2 turn a square into a constant and a directly integrable cosine. For example, cos² 2x = (1 + cos 4x)/2, so its primitive is x/2 + sin 4x/8 + C. The angle doubles as well as the original coefficient being halved. Differentiating this primitive gives (1 + cos 4x)/2 = cos² 2x.

Similarly, sin² 3x integrates to x/2 − sin 6x/12 + C. It is incorrect to use a power-rule guess such as sin³ 3x/3: differentiating that guess introduces both a cosine factor and an inner scale. The useful identity lets us use reverse differentiation without requiring the general substitution method in this section. For ∫cos² 2x from 0 to π/4, the sine term is zero at both endpoints, leaving exact area π/8.

Choose a relationship that gives standard derivative forms

Since sin 2x = 2 sin x cos x, a product sin x cos x can be replaced by sin 2x/2. Thus 2 sin x cos x integrates to −cos 2x/2 + C. The identity tan² u = sec² u − 1 gives ∫tan² 2x dx = tan 2x/2 − x + C on intervals with cos 2x ≠ 0. Rewriting does not remove any original tangent exclusions.

A non-negative integrand such as sin² x has equal signed integral and geometric area. Over [0, π], its primitive x/2 − sin 2x/4 gives π/2. A trigonometric product may change sign on another interval, so do not assume its definite integral is total area merely because an identity simplified it. Check the original sign and boundaries before giving a geometric or contextual interpretation.

Pause and explain

Which primitive integrates cos² 2x?

Put the idea to work

Worked example

Find ∫cos² 2x dx and evaluate its integral from 0 to π/4.

Show the worked solution

Use cos² 2x = (1 + cos 4x)/2.
Integrate the two terms to obtain x/2 + sin 4x/8 + C and verify by differentiation.
At both definite endpoints the sine term vanishes, giving π/8; the squared cosine is non-negative, so this is also area.

Answer x/2 + sin 4x/8 + C; exact integral and area on [0, π/4] are π/8.

Try a topic test

Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.

Test this topic →

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 0 to π of sin² x dx exactly.

Give me a hint

Rewrite the square as a constant minus a doubled-angle cosine.

Compare my reasoning

Use sin² x = (1 − cos 2x)/2.
A primitive is x/2 − sin 2x/4.
Both endpoint sine terms vanish, leaving π/2; the non-negative integrand makes this an area too.

π/2, also the geometric area on [0, π].

Look for these in your work

Use the sine-square identity with the correct sign.
Evaluate exact radian endpoints and check the integrand's sign.

2 · Independent

Find ∫tan² 2x dx and state the original restriction.

Give me a hint

Use tan² 2x = sec² 2x − 1 before integrating.

Compare my reasoning

The original tangent square requires cos 2x ≠ 0.
Integrate sec² 2x to tan 2x/2 and the constant −1 to −x.
Add C; differentiation gives sec² 2x − 1 = tan² 2x on each valid interval.

(1/2)tan 2x − x + C, with cos 2x ≠ 0.

Look for these in your work

Use a valid reciprocal identity before integrating.
Retain the original poles after checking the primitive.

3 · Transfer

A supplied velocity is v(t) = 2 sin t cos t cm/s for 0 ≤ t ≤ π/2 seconds, with radian phase. Find the displacement change.

Give me a hint

Use the double-angle product identity to recognise a standard sine integrand.

Compare my reasoning

Rewrite the velocity as sin 2t.
A displacement primitive is −cos 2t/2.
Endpoint subtraction gives 1/2 − (−1/2) = 1 cm.

Displacement change 1 cm.

Look for these in your work

Transform the original product using a correct identity.
Reverse the doubled angle and evaluate the signed displacement with units.

Common mistakes

A squared sine or cosine cannot be integrated by an unchanged power-rule guess.
Double the complete angle and reverse its new inner coefficient.
Keep tangent domain exclusions and distinguish signed accumulation from area.

Recall without your notes

What can you explain now?

Rewrite sin² 3x in a form with standard primitives.

Compare with the explanation

(1 − cos 6x)/2.

The sine-square identity doubles the inner angle 3x to 6x.

After trying it yourself, choose your next review. This is your self-assessment.

Your review choice appears on Today. Sign in to sync it across devices.

Continue to next lesson Back to Today

Make it stick

Choose an identity for a sine square, cosine square, sine-cosine product and tangent square. Differentiate each resulting primitive to check the inner factor and sign.

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.