Integration and Exact Techniques
Using antiderivatives, definite integrals, substitution, parts, partial fractions and standard trigonometric or exponential forms.
How to study A-level Mathematics
Define the objects and conditions, select a representation, execute a justified method, then validate and interpret the result.
Core concepts
Concept 1
Indefinite integration produces a family of antiderivatives and therefore requires a constant of integration.
Exam cue: Identify whether the integrand suggests reverse chain rule, parts or algebraic decomposition.
Concept 2
A definite integral represents signed accumulation and can be evaluated through an antiderivative.
Exam cue: Transform limits when using a substitution in a definite integral.
Concept 3
Substitution, integration by parts and partial fractions reverse different forms of differentiation or algebraic structure.
Exam cue: Differentiate an antiderivative to verify the integrand.
Risk pitfalls and guardrails
Omitting the constant from an indefinite integral.
Guardrail: Do not replace proof with examples, model conditions with unstated assumptions, or mathematical interpretation with unverified calculator output.
Treating signed area below the axis as positive geometric area.
Guardrail: Do not replace proof with examples, model conditions with unstated assumptions, or mathematical interpretation with unverified calculator output.
Changing variables but leaving incompatible limits or differentials.
Guardrail: Do not replace proof with examples, model conditions with unstated assumptions, or mathematical interpretation with unverified calculator output.
Memory anchors
Antiderivative
An antiderivative differentiates to the original function.
Definite Integral
A definite integral measures signed accumulation between its limits.
Substitution
Substitution changes variable to expose a simpler derivative pattern.
Integration by Parts
Integration by parts reverses the product rule.
Constant of Integration
An indefinite integral includes an arbitrary constant because constants differentiate to zero.
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
Find the general antiderivative of 6x² with respect to x.
Integrate 4x³−2x+5.
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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