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Source review · Integration techniques

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14 question records

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07 · Integration techniques

Edition metadata →

07-01 · Recognise the structure before integratingQuestion pp.1 / Solution pp.1, 2, 3

Question and explanations → · Complete native source article →

2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages. Printed/task labels inventoried: a, b, c, d, e, f, g, h, i, j.

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Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

$x>0$ .

On intervals where cos2θ≠0.

sin x≠0.

On intervals excluding ±4.

$x\ne-1/3$ .

$-1<x<1$ .

Exact private source page references

07 Integration tech.pdf · p.1
07 Integration tech.pdf · p.4
07 Integration tech sol.pdf · p.1
07 Integration tech sol.pdf · p.2
07 Integration tech sol.pdf · p.3

07-02 · Remove the modulus on the actual intervalQuestion pp.1 / Solution pp.3, 4

Question and explanations → · Complete native source article →

2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages. Printed/task labels inventoried: main.

Question / subparts

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2 page-item records.

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3 page-item records.

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1 page-item record. Native SVG generated; rendering is checked separately

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Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

No separate condition recorded. Check the full question, native working and printed assumptions.

Exact private source page references

07 Integration tech.pdf · p.1
07 Integration tech.pdf · p.4
07 Integration tech sol.pdf · p.3
07 Integration tech sol.pdf · p.4

07-03 · Change both the variable and its differentialQuestion pp.1 / Solution pp.4, 5

Question and explanations → · Complete native source article →

2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages. Printed/task labels inventoried: a, b.

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2 page-item records.

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3 page-item records.

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4 page-item records.

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Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Do not replace only 5ˣ by u while leaving dx unchanged. The factor $1/\ln5$ is essential.

Essential answer conditions and source qualifications

t≠±√3/2, on a connected interval.

Exact private source page references

07 Integration tech.pdf · p.1
07 Integration tech.pdf · p.4
07 Integration tech sol.pdf · p.4
07 Integration tech sol.pdf · p.5

07-04 · Use the supplied derivative, then split the signQuestion pp.1 / Solution pp.5

Question and explanations → · Complete native source article →

2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages. Printed/task labels inventoried: a.derivative, a.integral, b.

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2 page-item records.

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Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

a — integral

On intervals with cos(x²)≠0.

Exact private source page references

07 Integration tech.pdf · p.1
07 Integration tech.pdf · p.4
07 Integration tech sol.pdf · p.5

07-05 · Distinguish cos(x²) from cos²xQuestion pp.1 / Solution pp.6

Question and explanations → · Complete native source article →

2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages. Printed/task labels inventoried: a, b.indefinite, b.definite.

Question / subparts

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2 page-item records.

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2 page-item records.

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5 page-item records.

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0 page-item records.

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Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

No separate condition recorded. Check the full question, native working and printed assumptions.

Exact private source page references

07 Integration tech.pdf · p.1
07 Integration tech.pdf · p.4
07 Integration tech sol.pdf · p.6

07-06 · Build an inverse-sine integral from an earlier resultQuestion pp.1 / Solution pp.7

Question and explanations → · Complete native source article →

2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages. Printed/task labels inventoried: i, ii.

Question / subparts

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2 page-item records.

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Solution steps

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2 page-item records.

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Teacher notes

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4 page-item records.

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Tables

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Diagrams

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0 page-item records.

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

Differentiable working interval 0<x<1.

Exact private source page references

07 Integration tech.pdf · p.1
07 Integration tech.pdf · p.4
07 Integration tech sol.pdf · p.7

07-07 · Integrate twice and solve for the repeated integralQuestion pp.2 / Solution pp.8

Question and explanations → · Complete native source article →

2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages. Printed/task labels inventoried: main.

Question / subparts

Source comparison recorded

2 page-item records.

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Solution steps

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2 page-item records.

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Teacher notes

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2 page-item records.

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Tables

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0 page-item records.

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Diagrams

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0 page-item records.

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Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

The short answer key on question PDF p.4 (printed49) omits the factor $1/13$ and includes an unrelated part (a). The full solution PDF p.8 has the correct factor; the original key is preserved.

Essential answer conditions and source qualifications

Integral

The short key’s missing 1/13 is incorrect.

Exact private source page references

07 Integration tech.pdf · p.2
07 Integration tech.pdf · p.4
07 Integration tech sol.pdf · p.8

07-08 · Match a denominator derivative and remove a radicalQuestion pp.2 / Solution pp.8, 9

Question and explanations → · Complete native source article →

2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages. Printed/task labels inventoried: a.constants, a.integral, b.i, b.ii.

Question / subparts

Source comparison recorded

2 page-item records.

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Solution steps

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3 page-item records.

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Teacher notes

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3 page-item records.

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Tables

No item inventoried — owner check needed

0 page-item records.

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Diagrams

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

No separate condition recorded. Check the full question, native working and printed assumptions.

Exact private source page references

07 Integration tech.pdf · p.2
07 Integration tech.pdf · p.4
07 Integration tech sol.pdf · p.8
07 Integration tech sol.pdf · p.9

07-09 · Use the crossing before integrating an absolute valueQuestion pp.2 / Solution pp.10

Question and explanations → · Complete native source article →

2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages. Printed/task labels inventoried: a, b.sketch, b.integral.

Question / subparts

Source comparison recorded

2 page-item records.

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Solution steps

Source comparison recorded

2 page-item records.

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Teacher notes

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2 page-item records.

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Tables

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0 page-item records.

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Diagrams

Source comparison recorded

2 page-item records. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

No separate condition recorded. Check the full question, native working and printed assumptions.

Exact private source page references

07 Integration tech.pdf · p.2
07 Integration tech.pdf · p.5
07 Integration tech sol.pdf · p.10

07-10 · Choose the trigonometric branch before simplifying rootsQuestion pp.2 / Solution pp.11, 12

Question and explanations → · Complete native source article →

2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages. Printed/task labels inventoried: a, b.i, b.ii.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

3 page-item records.

Inspect native target → I compared this with the original

Teacher notes

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5 page-item records.

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Tables

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Diagrams

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

b(i) — integral

$x<1$ .

b(ii)

$0<x<1$ .

Exact private source page references

07 Integration tech.pdf · p.2
07 Integration tech.pdf · p.5
07 Integration tech sol.pdf · p.11
07 Integration tech sol.pdf · p.12

07-11 · Reuse two basic integrals and a radical substitutionQuestion pp.3 / Solution pp.13, 14

Question and explanations → · Complete native source article →

2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages. Printed/task labels inventoried: a.i, a.ii, a.iii, b.i, b.ii.

Question / subparts

Source comparison recorded

2 page-item records.

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Solution steps

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3 page-item records.

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Teacher notes

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4 page-item records.

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Tables

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0 page-item records.

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Diagrams

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

a(iii)

All a-integrands: 2−√3<x<2+√3.

b(i)

$x\ge0$ ; derivative on x>0.

Exact private source page references

07 Integration tech.pdf · p.3
07 Integration tech.pdf · p.5
07 Integration tech sol.pdf · p.12
07 Integration tech sol.pdf · p.13
07 Integration tech sol.pdf · p.14

07-12 · Turn the radical into a trigonometric identityQuestion pp.3 / Solution pp.14, 15

Question and explanations → · Complete native source article →

2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages. Printed/task labels inventoried: a, b.i, b.ii.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

3 page-item records.

Inspect native target → I compared this with the original

Teacher notes

Source comparison recorded

3 page-item records.

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Tables

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0 page-item records.

Inspect native target → I compared this with the original

Diagrams

Source comparison recorded

1 page-item record. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

The stated tan substitution uses x>0.

Exact private source page references

07 Integration tech.pdf · p.3
07 Integration tech.pdf · p.5
07 Integration tech sol.pdf · p.14
07 Integration tech sol.pdf · p.15

07-13 · Locate the crossing before taking the absolute areaQuestion pp.3 / Solution pp.15, 16

Question and explanations → · Complete native source article →

2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages. Printed/task labels inventoried: i, ii, iii.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

3 page-item records.

Inspect native target → I compared this with the original

Teacher notes

Source comparison recorded

3 page-item records.

Inspect native target → I compared this with the original

Tables

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Diagrams

Source comparison recorded

2 page-item records. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

No separate condition recorded. Check the full question, native working and printed assumptions.

Exact private source page references

07 Integration tech.pdf · p.3
07 Integration tech.pdf · p.5
07 Integration tech sol.pdf · p.15
07 Integration tech sol.pdf · p.16

07-14 · Check whether the given equality can holdQuestion pp.4 / Solution pp.16, 17

Question and explanations → · Complete native source article →

2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages. Printed/task labels inventoried: i, ii, iii.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

3 page-item records.

Inspect native target → I compared this with the original

Teacher notes

Source comparison recorded

3 page-item records.

Inspect native target → I compared this with the original

Tables

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Diagrams

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed with source inconsistency. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Confirmed source inconsistency: the equality in (iii) is impossible for the stated k. The printed constants are retained and explained separately; no replacement question or answer is invented. Question PDF p.4 (printed49), solution PDF pp.16–17.

Essential answer conditions and source qualifications

$-1<x<1$ .

iii

The left integral is positive but J<0. Formal rearrangement gives a=1/2,b=−3J/2≈7.94, but these are only a conditional identity with an impossible premise.

Exact private source page references

07 Integration tech.pdf · p.4
07 Integration tech.pdf · p.5
07 Integration tech sol.pdf · p.16
07 Integration tech sol.pdf · p.17

An empty omission list is not a completeness certificate. Add any issue you find to the article or page-item record before calling that part finished.