Power Integration
Integrate algebraic powers and powers of linear expressions, including the necessary coefficient adjustment.
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
reverse power rule
Exam cue: Can the integrand be rewritten as powers?
Concept 2
negative and fractional powers
Exam cue: Does a linear inner coefficient require division?
Concept 3
(x + q)ⁿ
Exam cue: Is the integral indefinite?
Concept 4
(px + q)ⁿ
Concept 5
constant of integration
Risk pitfalls and guardrails
Applying the power rule when n = −1
Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.
Forgetting to divide by the new exponent
Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.
Omitting + C from an indefinite integral
Guardrail: Use correct notation and check coefficients, signs and the constant of integration.
Memory anchors
Power up, divide
Increase the exponent by one and divide by that new exponent.
Minus one is excluded
The ordinary power rule does not apply to x⁻¹.
Inner linear factor divides
For (px + q)ⁿ, account for p in the denominator.
Rewrite before integrating
Use powers for roots, fractions and expanded terms.
Indefinite needs + C
All antiderivatives differ by a constant.
Differentiate to check
The derivative of the result should recover the integrand.
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
Integrate 6x² with respect to x.
Integrate 12x³−4x.
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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