PYQ Vault

Playbook

Definite Integration

Direct evaluation and piecewise integrands carry much of it; the a + b − x property is the time-saver when a direct attack looks hopeless.

Questions in the bank
196
q/paper in 2025–26
1.30
Numeric answer
33%
Notes pages
7

Tier: Core

When you’ll see it

An integral with limits, especially one that looks impossible directly, carries [x], {x} or |x|, has a variable limit, or is a limit of a long sum.

How this chapter is tested

Direct evaluation — one good substitution, parts or partial fractions — and piecewise integrands with [x], {x} or |x| carry much of the chapter. Piecewise questions are routine once the interval is split where the formula changes.

The property pages save the most time. When a direct attack looks hopeless, write the integral again with x → a + b − x and add the two forms. Over −a to a, drop the odd part; a denominator 1 + bˣ pairs with its reflection and halves the integral of an even numerator. Spotting the pattern is the whole skill.

Leibniz's rule turns an integral equation into a differential equation, which links the chapter to differential equations. Limits of sums become an integral through (1/n) Σ f(k/n). Reduction formulas and Beta integrals ask for a relation between members of a family, not a single value.

The sub-skills

The distinct skills inside the chapter, in the order to learn them.

  • Evaluating directly

    Substitute and change the limits; integrate by parts and work out the boundary term; split into partial fractions.

  • The a + b − x property

    ∫ from a to b of f(x) dx equals ∫ from a to b of f(a + b − x) dx; add the two forms. x → 1/x does the same on 1/a to a.

  • Odd, even and periodic integrands

    On −a to a odd parts vanish and even parts double; over n whole periods the integral is n times one period.

  • Greatest integer, modulus and max–min

    Split at the integers for [x], at the roots for |f(x)|, and where the two curves cross for max or min.

  • Leibniz's rule and integral equations

    d/dx of ∫ from g(x) to h(x) of f(t) dt is f(h)h′ − f(g)g′; an integral with constant limits is an unknown constant.

  • Reduction formulas and Beta integrals

    Link Iₙ to Iₙ₋₁ by parts; ∫ from 0 to 1 of x^(m − 1)(1 − x)^(n − 1) dx is B(m, n).

  • Limits of sums

    lim (1/n) Σ f(k/n) = ∫ from 0 to 1 of f(x) dx, with the upper limit set by the range of k.

Traps to expect

Distractor shapes this chapter reuses. The Traps page covers the ones that cut across chapters.

  • Old limits after a substitution

    With t = tan x, x = π/2 becomes t → ∞; with t = tan(x/2) it becomes t = 1. Keeping the old limits is the most common slip.

  • The greatest integer of a negative

    [−0.3] = −1, not 0. On an interval below zero, [x] is the next integer to the left.

  • The wrong period

    |sin x| repeats every π, not 2π. Using the longer period halves the number of copies and halves the answer.

  • The missing chain factor

    An upper limit x² brings a factor 2x in Leibniz's rule, and the lower limit enters with a minus sign.

Learn it before you drill it

This chapter has full teaching notes — foundations, worked examples, self-checks and a mastery check for each page. Read the notes once, then drill page by page below.

Definite Integration notes

Drill every Definite Integration question

196 questions from the bank, across 7 subtopics.

Drill the 196 questions

Drill one subtopic at a time

The 7 subtopics, in teaching order.

Related playbooks

Often paired with this one — the technique or the trap overlaps. Drill these next.