Playbook
Indefinite Integration
159 q - 3.35/paper - 51% HARD. A cornerstone you cannot skip and cannot rush. Substitution alone is 51 q; Trigonometric Integrals (Rational forms) runs 74% HARD, the highest of any subtopic in the subject. Foundations plus Trig Powers are 20 q at ~10% HARD - take those first.
- Questions in the bank
- 159
- q/paper (2024–25 shifts)
- 3.35
- Tagged HARD
- 51%
- Subtopics
- 6
Strand: Cornerstone
When you’ll see it
An antiderivative with no limits — and the immediate question of which of the four techniques (substitution, parts, partial fractions, trigonometric identity) the integrand is asking for.
How this chapter is tested
159 q at 3.35 per paper and 51% HARD. This is a cornerstone you can neither skip nor rush. It also contains the single hardest subtopic in the whole subject: Trigonometric Integrals - Rational and Substitution Forms, 35 q at 74% HARD. Nothing else in MHT-CET Maths runs that high.
There is a cheap corner, and it is small but real. Foundations and Standard Formulae (8 q, 13% HARD) plus Trigonometric Integrals - Powers and Identities (12 q, 8% HARD) come to 20 q at roughly a tenth the difficulty rate of the chapter as a whole. Take those two first: they are a fifth of a paper's integration marks for a fraction of the effort, and the standard-formula list is a prerequisite for everything else anyway.
The bulk is Integration by Substitution (51 q, 51% HARD), Rational Functions and Partial Fractions (27 q, 48%) and Integration by Parts (26 q, 54%). These are all recognition problems dressed as computation problems: the skill being tested is choosing the technique in the first fifteen seconds. A student who can classify an integrand quickly finishes this chapter comfortably; a student who tries substitution on everything runs out of clock.
This chapter is the clearest example of the no-negative-marking lever in the subject. The answer is a closed-form expression, so differentiating a candidate option is a legitimate and often much faster route than integrating the stem. When two techniques both look plausible and the clock is tight, differentiate rather than integrate.
The sub-skills
The distinct skills inside the chapter, in the order to learn them.
Foundations and standard formulae
The standard integral list, cold, including the inverse-trigonometric and logarithmic forms. 8 q at 13% HARD directly, and an unavoidable prerequisite for the other five subtopics.
Trigonometric integrals — powers and identities
Reduce powers and products of sine and cosine using double-angle and product-to-sum identities before integrating. 12 q at 8% HARD — the second-cheapest block in the chapter.
Integration by substitution
Spot that part of the integrand is the derivative of another part, substitute, and change the differential with it. 51 q at 51% HARD: the largest block, and the technique the other techniques fall back on.
Rational functions and partial fractions
Factor the denominator, decompose, and integrate term by term. Repeated factors and irreducible quadratics each need their own decomposition shape. 27 q at 48% HARD.
Integration by parts
Choose which factor to differentiate and which to integrate, then apply the formula — and recognise the cases that return to the original integral and are solved by rearrangement. 26 q at 54% HARD.
Trigonometric integrals — rational and substitution forms
Integrands that are rational in sine and cosine, handled by the half-angle substitution or by splitting the numerator to match the denominator's derivative. 35 q at 74% HARD, the highest of any subtopic in the subject. Learn it last and expect it to cost real time.
Traps to expect
Distractor shapes this chapter reuses. The Traps page covers the patterns that cut across chapters.
Substitution without changing the differential
Replacing the expression but keeping the original dx. The resulting answer is off by a factor and that factor is exactly what one distractor uses.
Missing modulus inside a logarithm
Standard logarithmic antiderivatives carry an absolute value. Options identical apart from the modulus are common, and the version without it is the trap.
Partial fractions with the wrong decomposition shape
A repeated linear factor needs a term for each power, and an irreducible quadratic needs a linear numerator. Using the simple shape produces a decomposition that cannot reproduce the original fraction.
Antiderivative of a lookalike integrand
A distractor that is a perfectly valid antiderivative — of a slightly different function. Differentiating it and comparing with the stem exposes it in seconds.
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.
Indefinite Integration notesDrill every indefinite integration question
159 questions from the bank, scoped to 6 bundled subtopics.
Drill one subtopic at a time
The 6 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.