Exams
Topic module

Trigonometric Integration and Differential Equations

Use the supplied standard trig integrals with coefficient adjustments and integrate dy/dx = f(x).

Long-form learning
Concept to Risk to Memory to Check-up

How to study Higher Mathematics

Practise a skill without a calculator first, connect it to neighbouring techniques, then apply it in unfamiliar contexts with complete working and interpretation.

Core concepts

Concept 1

integral of sin ax

Exam cue: What coefficient appears inside the angle?

Concept 2

integral of cos ax

Exam cue: Which sign belongs to the sine antiderivative?

Concept 3

phase-shifted trig

Exam cue: Is an initial condition available to determine C?

Concept 4

dy/dx = f(x)

Concept 5

integration constant

Risk pitfalls and guardrails

Using derivative coefficients instead of reciprocal integral coefficients

Guardrail: Use correct notation and check coefficients, signs and the constant of integration.

Missing the negative sign when integrating sine

Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.

Stopping with y plus an undetermined constant when a condition is given

Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.

Memory anchors

Integral sine is negative cosine

∫sin(ax)dx = −cos(ax)/a + C.

Integral cosine is positive sine

∫cos(ax)dx = sin(ax)/a + C.

Inside coefficient divides

Reverse differentiation by dividing by the angle multiplier.

Shift stays inside

Preserve qx + r while adjusting for q.

dy/dx invites integration

Integrate f(x) to recover the family of y-functions.

Condition fixes C

Substitute the known point after integrating.

Checkpoint rule

Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.

Knowledge Check (after reading)

Short check-up to confirm understanding of this module.

Check-up Questions

1-2 question checkpoint

Integrate cos(4x) with respect to x.

Integrate sin(3x) with respect to x.

Answer all questions to submit.

Next step personalized recommendations

Continue learning

Move forward only after this module is stable.

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