Playbook
Definite Integration
73 q - 1.85/paper - 45% HARD. Two subtopics. Symmetry, King's Property and Absolute Value (42 q, 38% HARD) is the one to own: those questions collapse in a line once you spot the property, which is the highest-leverage recognition in the calculus block.
- Questions in the bank
- 73
- q/paper (2024–25 shifts)
- 1.85
- Tagged HARD
- 45%
- Subtopics
- 2
Strand: Long Tail
When you’ll see it
An integral carrying numeric limits — especially symmetric limits, limits running 0 to a, or an absolute value or piecewise expression inside.
How this chapter is tested
73 q, 1.85/paper, 45% HARD, and the chapter splits into a recognition half and a grind half. Symmetry, King's Property and Absolute Value is 42 q at 38% HARD; Substitution and Standard Form is 31 q at 55%.
The 42-q half is the highest-leverage recognition anywhere in the calculus block. Once the property is spotted the question collapses in a single line — an odd integrand over symmetric limits is zero with no antiderivative computed at all, and King's property turns an unintegrable-looking expression into twice something trivial or into a constant. At 1.8 minutes a question, that is worth more than the two marks it scores.
The other half is ordinary integration with limits attached, which makes it Indefinite Integration (159 q, 3.35/paper, 51% HARD) plus one extra step. Its 55% HARD rate is real, but so is the transfer: everything invested in the cornerstone integration chapter is paid back here, so this half needs almost no separate preparation.
Order of attack follows directly: scan every definite integral for a property BEFORE reaching for a technique. Ten seconds of looking saves a minute of integrating on roughly three of every five questions in this chapter.
The sub-skills
The distinct skills inside the chapter, in the order to learn them.
Standard-form evaluation with limits
Integrate, then evaluate at the upper limit minus the lower. When you substitute, change the limits to the new variable rather than back-substituting at the end — it is faster and removes a whole class of error.
Even and odd over symmetric limits
Over -a to a, an odd integrand gives zero and an even integrand gives twice the integral from 0 to a. Test the parity of the integrand first, before anything else.
King's property
The integral from a to b of f(x) equals the integral of f(a + b - x). Adding the two forms usually cancels the awkward part and leaves a constant times the length of the interval.
Absolute value and piecewise integrands
Find where the expression inside changes sign, split the interval there, and integrate each piece with the correct sign. A modulus integrated as if it were the bare expression is the commonest wrong answer.
Periodicity and interval-shifting properties
For a periodic integrand, the integral over a whole number of periods reduces to a multiple of the integral over one period, and the starting point of the interval does not matter.
Traps to expect
Distractor shapes this chapter reuses. The Traps page covers the patterns that cut across chapters.
Substituting without changing limits
The limits belong to the original variable. Evaluating the new antiderivative at the old limits gives a plausible number, and it is on the option list.
Odd integrand, asymmetric limits
The zero shortcut needs limits of the form -a to a. Applying it to 0 to a, or to -a to 2a, produces zero where the answer is not zero.
Signed integral offered as area
Where the integrand dips below the axis, the integral and the area differ. If the question says area, split at the crossing and take magnitudes.
Swapped limits
Reversing the limits negates the integral. A distractor equal to the correct answer with the opposite sign usually means exactly this.
Drill every definite integration question
73 questions from the bank, scoped to 2 bundled subtopics.
Drill one subtopic at a time
The 2 subtopics this playbook covers, in catalog order.
Related playbooks
Often paired with this one — the technique, the trap or the taxonomy overlaps. Drill these next.