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Definite Integration

67 q - 1.75/paper - 46% HARD. Five pages: the evaluation toolkit (standard forms, substitution with changed limits, by parts), the trigonometric block (tan x = t, half-angle forms), then the three properties that collapse a question in a line - odd/even symmetry, King's property with the f/(f + g) family, and modulus/greatest-integer splitting. The property pages are 43 of the 69 q and the highest-leverage recognition in the calculus block.

Questions in the bank
67
q/paper (2024–25 shifts)
1.75
Tagged HARD
46%
Subtopics
5

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

67 q, 1.75/paper, 46% HARD, and the chapter splits into a recognition half and a grind half. The three property pages — odd and even symmetry, King's property, modulus and greatest-integer splitting — are 41 q at 39% HARD; the two evaluation pages are 26 q at 58%.

The 43-q property block 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 (151 q, 3.42/paper, 52% HARD) plus one extra step. Its 58% 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.

  • Evaluating Definite Integrals — Standard Forms, Algebraic Substitution and By Parts

    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; split a numerator against a quadratic; use by parts for inverse trig and e^x (f + f'), and the reduction I_n + I_(n-2) = 1/(n-1) for powers of tan.

  • Trigonometric Definite Integrals — tan x = t, Half-Angle Forms and Powers

    Divide by a power of cos x and put tan x = t (limits 0 to 1 at pi/4, 1/sqrt3 at pi/6); 1 + cos x = 2 cos^2(x/2); the Weierstrass result for 1/(a + b cos x) over 0 to pi is pi/sqrt(a^2 - b^2). The 73% HARD corner of the chapter.

  • Odd and Even Integrands — 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 first, split a mixed integrand into its odd and even parts, and shift the variable when the interval is symmetric about a point other than 0.

  • King's Property — f(a + b − x) and the f/(f + g) Family

    The integral from a to b of f(x) equals the integral of f(a + b - x). Adding the two forms cancels the awkward part: f/(f + g) over a to b is (b - a)/2, x f(sin x) over 0 to pi is (pi/2) times the integral of f(sin x), and f(x)/(1 + e^x) over -a to a is the integral of f over 0 to a.

  • Modulus and Greatest-Integer Integrands — Split the Interval

    Find where the expression inside changes sign or where [x] steps, split the interval there, and integrate each piece with its own sign or constant. A modulus integrated as if it were the bare expression is the commonest wrong answer.

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.

Learn it before you drill it

This chapter has full teaching notes — foundations, worked examples, self-checks and a per-subtopic mastery checkpoint. Read the notes once, then drill subtopic by subtopic below.

Definite Integration notes

Drill every definite integration question

67 questions from the bank, scoped to 5 bundled subtopics.

Drill the 67 questions

Drill one subtopic at a time

The 5 subtopics this playbook covers, in catalog order.

  • Evaluating Definite Integrals — Standard Forms, Algebraic Substitution and By PartsDrill
  • Trigonometric Definite Integrals — tan x = t, Half-Angle Forms and PowersDrill
  • Odd and Even Integrands — Symmetric LimitsDrill
  • King's Property — f(a + b − x) and the f/(f + g) FamilyDrill
  • Modulus and Greatest-Integer Integrands — Split the IntervalDrill

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