The two properties that let an integral be broken apart and rescaled

What to assume they know

• The anti derivative and the family it belongs to, from the first topic of this module
• The list of standard formulae, from the second topic of this module
• The sum rule and the constant-multiple rule for derivatives, from Chapter 5 of Part I
• Set-builder notation, and what it means for two sets to be identical
• The fact that a function with zero derivative throughout an interval is constant

What they should be able to do

• List the five properties the section states and say which two are used in practice and which two are used to prove them
• State the two halves of the inverse relationship separately, and say why only one of them carries a constant
• Explain what it means for two indefinite integrals to be equivalent, in terms of the two families they name
• Read the Note on Part II p. 230 and say what an equals sign between two indefinite integrals is asserting
• Reproduce the chapter's proof that an integral distributes across a sum, and identify which earlier property each step uses
• Take a constant factor out of an integral, and say what the statement means when the constant is zero
• Apply the combined statement to a finite sum of scaled functions
• Justify writing a single constant of integration in a final answer where several arose during the work
• Say what these properties do not license, and give the standard wrong inference

Where it usually goes wrong

• "The integral of a product is the product of the integrals." Nothing in this section says anything about products. The two properties are about sums and about constant factors, and the constant has to be constant. Section 11 exists because this is the single most common error the section invites, and because the technique that does handle products is five sections away.
• "A constant factor can come out, so a function factor can too." Property (IV) names a real number. A factor that varies with the variable of integration is part of the integrand and stays there.
• "Property (I) says integration and differentiation cancel, full stop." It says so in two statements that do not match: one returns the integrand exactly, the other returns the function plus a constant. Students who collapse the two into one slogan lose the constant permanently.
• "Two integrals joined by an equals sign are two equal numbers." They are two families, and the Note on Part II p. 230 says so. This is why an answer differing from the printed one by a constant is not wrong, and students who do not know it spend the year believing they have made errors they have not.
• "Writing one constant instead of two is sloppiness the book tolerates." It is a stated convention adopted at a stated place, and it is legitimate because the difference of two arbitrary constants is another arbitrary constant. Example 2 (i) performs the merge in full before the convention is adopted, which is exactly the right order.
• "Property (V) is a new result." It is (III) and (IV) applied repeatedly, and the chapter says so rather than proving it. Presenting it as a separate fact makes the section look like five things to remember instead of two.
• "Because the Summary lists only two properties, the other three do not matter." The two the Summary keeps are the two that get used; the two it drops are the two that make them true. A student revising from Part II p. 287 alone keeps the tools and loses the reason they work.
• "Zero times an integral is zero, so the integral of the zero function is zero." The integral of the zero function is the family of all constant functions. This is a genuine wrinkle and the Note on Part II p. 230 is what resolves it; it is not a flaw in Property (IV).

Questions to check understanding

• State the two properties that are used in practice, and identify which of them a given step of working relies on
• Explain why differentiating an integral and integrating a derivative give answers of different shapes
• Prove that an integral distributes across a sum, using the chapter's own route through Property (II)
• Integrate a finite sum of scaled standard functions in one line — the form of Exercise 7.1 Q8, Q9, Q16 and Q17
• Justify replacing two constants of integration by one
• Say whether a proposed manipulation is licensed, given a product or a quotient in the integrand
• Explain what an equals sign between two indefinite integrals asserts

Examples worth working on the board

Values marked verified are worked out here from the chapter's own printed data; neither answers file was opened, and this chapter prints no answers to its exercises.

• The five properties as a set (§7.2.1, Part II pp. 229–231). They are labelled with roman numerals (I) to (V) and four of them carry a proof; (V) is asserted as a generalisation of the two before it. Read off the page images. The explanation's first section should show all five at once and say plainly that only (III), (IV) and (V) are ever used on an exercise — (I) and (II) exist to prove them.
• Property (I) (Part II p. 229). Two displayed statements. Differentiating an indefinite integral returns the integrand with nothing attached; integrating a derivative returns the function plus a constant. Verified as asymmetric on purpose: the first composes derivative-after-integral, and the constant dies under the derivative; the second composes integral-after-derivative, and the constant survives because the integral introduces it. The chapter proves both in four lines and does not comment on the asymmetry. Section 2 is exactly that comment.
• Property (II) and its proof (Part II pp. 229–230). If two indefinite integrals have the same derivative, their difference has zero derivative and so is constant; the two families they name are then identical. The chapter marks the step where the constant appears with a bracketed prompt asking the reader why — answer it: it is the previous topic's Remark, applied to the difference. Read off the page image.
• The boxed Note (Part II p. 230). It says that writing one indefinite integral equal to another is the customary way of asserting that the two families coincide, with the parameter left unmentioned. This is the most under-read sentence in the section. It means that an equals sign between two indefinite integrals is a statement about two sets, not about two numbers, which is why the constants can be shuffled, merged and renamed without anyone objecting. Give it a section of its own.
• Property (III) and its proof (Part II p. 230). Differentiate the integral of a sum and get the sum; differentiate the sum of the two integrals and get the same sum; conclude by Property (II) that the two are equivalent. Three steps: never compare integrals directly, compare their derivatives and appeal to (II).
• Property (IV) and its proof (Part II pp. 230–231). Same template, one line shorter. A constant factor passes through the integral sign. An observation the chapter does not make: at a zero factor the left side is the integral of the zero function, which is the family of all constants, and the right side is zero times a family. The Note on Part II p. 230 is what makes that an acceptable identity rather than a contradiction, and it is worth thirty seconds because it is the sharpest illustration of what the Note licenses. Flag it as an added remark.
• Property (V) (Part II p. 231). The two preceding statements combined over a finite list of functions and a finite list of real coefficients. Stated without proof, since it follows by repetition. This is the form actually used in practice and the form the Summary keeps.
• Example 2 (i) (Part II p. 232). A cube less one, all over a square, split into the variable and a negative power, integrated separately, and the two constants written out as separate symbols before being combined into a single one. The chapter names Property (V) in the margin of the first step. Verified: the answer is half a square plus the reciprocal of the variable, plus one constant, and it differentiates back to the original integrand. This is the only place in the chapter where two constants are carried visibly and then merged, and it is the whole justification for the habit that follows.
• The Note after Example 2 (i) (Part II p. 232). One sentence adopting the convention of a single constant in every final answer from that point on. Everything after Part II p. 232 in this chapter relies on it silently.
• Example 3 (Part II p. 233), three parts, each a sum split by Property (III) after a rewrite. Verified: (i) a sine plus a cosine integrates to minus the cosine plus the sine; (ii) the cosecant times the sum of the cosecant and the cotangent expands to two standard rows and integrates to minus the cotangent less the cosecant; (iii) one less a sine over the square of the cosine splits into the square of the secant less the secant times the tangent and integrates to the tangent less the secant. Use part (ii) for section 5: the split is invisible until the bracket is expanded, and the property is doing the work that makes expanding worthwhile.
• Exercise 7.1, questions 6 to 20 (Part II pp. 234–235). Every one of the fifteen is Property (III) or (IV) or both, applied after a rewrite. Q8, Q9, Q15, Q16 and Q17 are the cleanest illustrations of a coefficient coming out front, and Q6 is the shortest.
• The Summary's property bullet (Part II p. 287). Read off the page image. It gives two numbered properties — the sum and the constant factor — and then the generalisation over a finite list. Properties (I) and (II) do not appear. See section 10 and the note below.

Figures to have open

• Two stacks of vertically translated curves for section 3, drawn to the same scale so that showing them identical is a visual claim rather than an assertion. Not in the book; the chapter draws no picture in this section and speaks of families of curves in words only, on Part II p. 230.
• A five-row table of the properties for section 1, and a two-column comparison against the Summary's list for section 10. Both are the chapter's own content, from Part II pp. 229–231 and Part II p. 287; the layout is added here. Build both with the repo's DataTable component.
• A step-by-step migration of a constant factor across the integral sign for section 6. Not in the book.
• No figure is available from the chapter for any section of this topic. The whole chapter prints one figure, on Part II p. 267, and it belongs to module m03. Every picture listed here is an added construction over the chapter's own content.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 7 "Integrals", §7.2.1 Some properties of indefinite integral, Properties (I) and (II) with proofs, Part II pp. 229–230
• The boxed Note on the suppressed parameter, Part II p. 230
• Properties (III), (IV) and (V) with proofs, Part II pp. 230–231
• Example 2 (i) and the Note adopting a single constant, Part II p. 232
• Example 3, Part II p. 233; Exercise 7.1, questions 6 to 20, Part II pp. 234–235
• Summary, the indefinite-integral properties bullet, Part II p. 287