Asking which function had this derivative, and why the answer is a whole family

What to assume they know

• The derivative of a function of one real variable, from Chapter 5 of Part I
• The derivatives of the six trigonometric ratios, the exponential, the natural logarithm and a general power, all as a memorised list
• What it means for a derivative to vanish everywhere on an interval
• Interval notation, and the phrase for every point of an interval
• Set-builder notation, and reading a set whose members are indexed by a parameter
• The two inverse trigonometric functions whose derivatives appear in the standard list, from Chapter 2 of Part I

What they should be able to do

• State the question integral calculus is set up to answer, in terms of a given derivative and an unknown function
• Name the two problems the chapter's opening page identifies, and match each to one of the two kinds of integral
• Produce an anti derivative for a stated function by recognising it as the derivative of something already known
• Explain why adding any real constant to an anti derivative produces another one, from the derivative of a constant
• Reproduce the chapter's argument that two functions with equal derivatives on an interval differ by a constant, and say why that argument completes the answer rather than merely extending it
• Read the integral sign as a name for an entire class of functions rather than for one function
• Use the chapter's vocabulary table to name the parts of an integral expression
• Recover the single member of a family that satisfies one prescribed value, and say why one condition is exactly enough
• Give the chapter's own instance of a function with no anti derivative among the familiar ones, and explain what is and is not being claimed
• Rewrite a standard integral formula in a variable other than the usual one, and say why nothing changes

Where it usually goes wrong

• "The constant is a formality you tack on at the end." It is the whole content of this topic. Without the Part II p. 227 Remark the constant is an observation; with it, the constant is a complete description of every possible answer. An explanation that treats it as bookkeeping has skipped the theorem.
• "Every anti derivative of a function differs from a given one by a constant — that's obvious." It is not obvious, and it is false without the hypothesis that the domain is an interval. The chapter states the Remark on an interval and the argument uses that. The explanation may keep the interval quietly in place, but it must not present the conclusion as a definition.
• "Integration is just differentiation done backwards, so the same rules apply in reverse." The inverse relationship holds statement by statement, not rule by rule. There is a product rule for derivatives; the corresponding statement for integrals is a technique that sometimes helps and sometimes does not, and it arrives four sections later. Students who expect symmetry here are set up to invent a quotient rule for integrals.
• "The exponential of a negative square has no anti derivative." The chapter says something weaker and more careful: none of the familiar named functions has it as a derivative, so inspection will not find one.
• "Adding a constant gives a different function, so the answer is ambiguous and therefore useless." The family is the answer, and every application that needs a single function supplies one extra fact that names it — which is what Example 4 does. Ambiguity here is structure, not defect.
• "An anti derivative and an integral are two different things." In this chapter's vocabulary they are the same object under two names, and Table 7.1 says so in the row that glosses an integral of a function. The distinction students are reaching for is between the indefinite integral and the definite integral, and that is module m03.
• **"The letter under the d carries meaning, so changing it changes the answer."** Remark (iii) exists precisely to say otherwise. The letter says which symbol is being varied; nothing else about the statement depends on it.
• "The portrait means he invented this." The chapter prints a portrait, a pair of dates and nothing else. It never says what he did, and it names no second person. Anything the explanation adds about the history is added here and must be sourced elsewhere.

Questions to check understanding

• Write an anti derivative of a stated function by inspection, and verify it by differentiating
• Explain why an anti derivative is never unique, and state precisely how much freedom there is
• Reproduce the argument that two functions with equal derivatives on an interval differ by a constant
• Find the member of a family that takes a prescribed value at a prescribed point — the form of Example 4 and of Exercise 7.1 Q22
• Name the parts of a written integral expression, given the expression
• Choose the correct anti derivative from four options — the form of Exercise 7.1 Q21 and Q22
• Rewrite a standard integral formula in a different variable and justify that the content is unchanged
• State what the chapter does and does not claim about the exponential of a negative square

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 opening two paragraphs (§7.1, Part II pp. 225–226). They set differential calculus against integral calculus by what motivated each: tangent lines and slopes on one side, the area of a region under a graph on the other. They then ask the reverse question — given the derivative at every point of an interval, can the function be recovered — and name the candidates. Everything in this topic unpacks that reverse question.
• The two listed problems (§7.1, Part II p. 226). The page numbers them (a) and (b): recovering a function from its derivative, and finding an area bounded by a graph under stated conditions. It then says these produce the two kinds of integral. Use it as the map for the whole chapter: module m01 and m02 are problem (a); module m03 is where problem (b) is answered.
• The three opening derivatives (§7.2, Part II p. 226), numbered (1), (2) and (3): the derivative of the sine is the cosine; the derivative of a cube divided by three is the square; the derivative of the exponential is itself. Read off the page image. The chapter then reruns all three with a constant added and observes that nothing changes. Verified: each of the three is a Class XI derivative and the constant contributes zero, so all three restatements hold.
• The Remark on equal derivatives (§7.2, Part II p. 227). Two functions with the same derivative on an interval are subtracted; the difference has zero derivative throughout; a function whose rate of change is zero on an interval is constant; therefore the two differ by a constant. Four steps. Verified: this is the only thing in the section that earns the word all — without it, adding constants merely produces more anti derivatives, and there could be others of a different shape.
• The introduction of the symbol (§7.2, Part II p. 227). The chapter says outright that the new symbol stands for the entire class, not for one function, and then writes the class as one anti derivative plus a constant. It also gives a Notation line converting a statement about a derivative into a statement about an integral. Both belong in the explanation in section 6.
• Table 7.1 (Part II p. 227). Seven rows, two columns headed for the symbol or phrase and for its meaning. Read off the page image, the rows are: the integral of a function with respect to a variable; the integrand; the variable of integration; integrate as an instruction; an integral of a function, glossed as one that differentiates back to it; integration as the process; and the constant of integration, glossed as any real number treated as a constant function. Build it with the repo's DataTable. This is a glossary, not a set of rules.
• Example 4 (Part II pp. 233–234). The function is four times a cube, less six, and the anti derivative is required to take the value three at zero. The general anti derivative is a fourth power less six times the variable, plus a constant; substituting zero forces the constant to be three. Verified: differentiating a fourth power less six times the variable returns four times a cube less six, and at zero the expression is the constant alone, so the prescribed value fixes it. This is the cleanest illustration in the chapter of one condition selecting one member.
• Remark (i) (Part II p. 234). It says what Example 4 did in general terms: knowing one anti derivative hands you infinitely many, and an extra condition is often what an application supplies to cut back to one. Note that the chapter does not use the phrase initial condition.
• Remark (ii) (Part II p. 234). The chapter names a specific integrand — the exponential of the negative of a square — and says it cannot be done by inspection because no familiar function has that derivative. Read off the page image; the exponent is negative and the square is of the variable, and both matter. Say carefully what is claimed: the chapter's sentence is about the familiar named functions and their inverses.
• Remark (iii) (Part II p. 234). The power rule restated with a different letter, to make the point that the formulas are modified to match whatever variable is in use. Two lines is enough.
• Exercise 7.1, questions 1 to 5 (Part II p. 234). Anti derivatives by inspection: the sine of twice the variable; the cosine of three times it; the exponential of twice it; a linear expression squared; and a difference of a sine and a multiple of an exponential. Verified, working added here: halve and take minus the cosine of twice it; take a third of the sine of three times it; halve the exponential of twice it; take the cube of the linear expression divided by three times its leading coefficient; and combine the first of these with a third of four times the exponential of three times the variable. Each is checked by differentiating back. The chapter prints no answers.
• Exercise 7.1, question 22 (Part II p. 235). Four options for the function that differentiates to four times a cube less three over a fourth power and vanishes at two. Verified by working added here: integrating gives a fourth power plus the reciprocal of a cube, plus a constant; at two that is sixteen plus an eighth plus the constant, so the constant is minus one hundred twenty-nine over eight, and the first option is the one that matches. It belongs here rather than in the next topic because it is Example 4's shape with an exercise's numbers.
• The Summary's first bullet (Part II p. 287). It restates the inverse relationship in words, gives the symbol, names the result an indefinite or general integral, and closes by saying the members of the family differ by a constant. It is a faithful condensation of this topic and can be used as the closing card.
• The chapter frontispiece (Part II p. 225). A QR code carrying the Part II catalogue number and the chapter number, an epigraph attributed to James B. Bristol about studying new material, and an engraved portrait captioned with initials, the surname Leibnitz and the dates 1646 to 1716. Caption these on a card; do not reproduce the page. See the two notes below before the name is shown.

Figures to have open

• A stack of vertically translated copies of one curve, all with the same slope at every abscissa, for section 6. The chapter draws nothing of the kind and the device is added here; it is the single most useful picture in the topic and it should be the same drawing that reappears in section 8 with one member picked out.
• A two-column table of the seven Table 7.1 rows for section 7. The content is the chapter's own Part II p. 227 table; build it with the repo's DataTable component.
• A backwards arrow diagram for section 1: a named function on the right, its derivative on the left, and the query arrow drawn from left to right. Not in the book.
• The Leibnitz portrait and the Bristol epigraph on Part II p. 225 are the textbook's own. Use a caption card naming the portrait and its dates, and read the notes below on the printed spacing of the initials. Ignore the QR code.
• No figure is needed for sections 2, 3, 4, 5, 9 or 10, and none is available: the whole chapter prints exactly one figure, on Part II p. 267, and it belongs to a different module. See Notes for how that was established.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 7 "Integrals", §7.1 Introduction, Part II pp. 225–226
• §7.2 Integration as an Inverse Process of Differentiation, the three opening derivatives and the arbitrary constant, Part II p. 226
• The Remark on functions with equal derivatives, the integral symbol and the Notation line, and Table 7.1, Part II p. 227
• Example 4 and Remarks (i) to (iii), Part II pp. 233–234
• Exercise 7.1, questions 1 to 5 and question 22, Part II pp. 234–235
• Summary, the first bullet, Part II p. 287