Reading the table of standard integrals off the table of derivatives

What to assume they know

• The derivative of a general power, of the six trigonometric ratios, of the exponential and of the natural logarithm, from Chapter 5 of Part I
• The derivatives of the inverse sine, inverse cosine and inverse tangent, from Chapter 5 of Part I
• What an anti derivative is and why it comes with a constant, from the previous topic
• Rewriting a trigonometric expression using the Pythagorean identities and the definitions of the four derived ratios
• Splitting an algebraic fraction term by term when the denominator is a single power
• Index laws, including negative and fractional exponents

What they should be able to do

• Read any row of the chapter's two-column list in both directions and say which direction is the definition and which is the consequence
• State the exponent the power rule excludes, and name the row that covers the excluded case
• Reproduce the six trigonometric rows with their signs correct, and say where each minus sign comes from
• Explain why two of the rows have the same left-hand side and different right-hand sides, and why that is not a contradiction
• Say why the logarithm row is written with a modulus and what would go wrong without it
• Integrate a general exponential, and recover the natural case as a special one
• State what the Note on Part II p. 229 suspends and when a solver has to reinstate it
• Guess an anti derivative for a composed function and correct it by a constant factor, checking by differentiating
• Rewrite an integrand that is not in the list until every piece of it is
• Compare the chapter's own list against the Summary's and account for the difference

Where it usually goes wrong

• "There is no formula for the integral of the reciprocal of the variable, because the power rule excludes it." The power rule excludes it and row (xii) supplies it. The two live ten rows apart on facing pages, which is exactly why students conclude the case is missing.
• "Two integrals of the same function must be equal, so one of rows (viii) and (ix) is wrong." Neither is wrong. Their answers differ by a constant, which the previous topic's Remark says is the only way two anti derivatives can differ. Students who have not internalised that read the pair as a misprint.
• "The modulus in the logarithm row is a convention." It is what makes the row true for negative arguments, and Example 1 (iii) derives it by running the two sign cases separately. Drop the bars and the formula asserts the logarithm of a negative number.
• "The list is closed — anything not on it cannot be integrated." The list is the reversal of a derivative list and nothing more. Four trigonometric integrals arrive in §7.3.1, six quadratic forms in §7.4, and three surd forms in §7.6.2, none of which are here. Section 11 exists to say so and to point at each.
• "Because the derivative rows are correct, no domain needs checking." The Note on Part II p. 229 says the opposite: the interval is normally suppressed and has to be reinstated when a specific problem needs it. A student who integrates the reciprocal of the variable across zero has ignored it.
• "Halving the answer for the cosine of twice the variable is a trick." It is the correction forced by differentiating the guess and finding a spare factor. Say it as a two-step procedure — guess, differentiate, divide by whatever appeared — and it stops being a trick and starts being the substitution method of the next module in embryo.
• "The general-base row needs a separate rule when the base is the exponential's own." It does not; the divisor becomes one and the row collapses to the exponential row. Show the collapse rather than asserting it.
• "Every trigonometric integrand needs an identity." Only those not already in the list. Example 3 (i) is two rows read straight off; part (iii) needs one algebraic split and no identity at all.

Questions to check understanding

• Write down the integral of a stated standard function from memory, with the constant and any modulus in place
• Name the exponent the power rule excludes and give the formula that covers it
• Explain why one integrand can have two printed answers, and compute their difference
• Integrate a general exponential with a stated base
• Find an anti derivative by inspection and verify it by differentiating — the form of Exercise 7.1 Q1 to Q5
• Rewrite an integrand into pieces that are all standard, then integrate — the form of Exercise 7.1 Q6 to Q20
• Choose the correct anti derivative from four options — the form of Exercise 7.1 Q21
• State the interval on which a stated standard formula is valid, given that the chapter suppresses it

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 two-column list (§7.2, Part II pp. 228–229). Read cell by cell off the printed page images. The left column is headed for derivatives, the right for integrals with anti derivatives in brackets, and there are thirteen numbered rows, (i) through (xiii), the first of which carries a second, unnumbered line for the constant integrand. In order: a general power with the exclusion attached; the cosine; the sine with its minus sign on the answer; the square of the secant; the square of the cosecant with its minus sign; the secant times the tangent; the cosecant times the cotangent with its minus sign; the reciprocal of the square root of one less a square, answering with the inverse sine; the same integrand answering with minus the inverse cosine; the reciprocal of one plus a square, answering with the inverse tangent; the exponential; the reciprocal of the variable, answering with the logarithm of the modulus; and a general base raised to the variable, divided by the logarithm of that base. Every minus sign and every modulus bar above was read at 100 dots per inch from the page images.
• The exclusion on row (i) (Part II p. 228). The condition sits to the right of the formula and rules out one exponent. Verified: at that exponent the denominator of the answer would vanish, and the case is not lost — row (xii) is exactly it. Show the two rows adjacent; the chapter separates them by ten rows and a page break, which is the single likeliest place for a student to conclude the case is missing.
• The pair of rows with one integrand (rows (viii) and (ix), Part II p. 228). Both integrate the reciprocal of the square root of one less a square; one answers with the inverse sine, the other with minus the inverse cosine. Verified: the two answers differ by a constant — the inverse sine and the inverse cosine of the same argument add to a fixed value — so both are anti derivatives and the previous topic's Remark says they must differ by a constant, which they do. This is the best evidence in the chapter that the constant is doing real work rather than decorating the answer. The chapter prints the pair without comment.
• The Note (Part II p. 229). One boxed sentence saying the interval on which each function is defined is normally left unstated, and that a specific problem obliges the solver to keep it in view. Verified as load-bearing: row (viii) needs the argument strictly between minus one and one; row (xii) needs the variable non-zero; the trigonometric rows need the points where the ratio blows up removed. None of that is printed beside the rows.
• The method-of-inspection paragraph (§7.2.1, Part II p. 231). It says the search is for a function that differentiates to the given one, names the approach, and hands off to examples. Two sentences.
• Example 1 (Part II pp. 231–232), three parts. (i) The cosine of twice the variable: differentiating the sine of twice the variable overshoots by a factor of two, so halve it. (ii) Three times a square plus four times a cube: the cube plus the fourth power differentiates to exactly that, no correction needed. (iii) The reciprocal of the variable: the chapter runs the positive and negative cases separately and then combines them into the logarithm of the modulus. Verified: all three differentiate back. Part (iii) is where the modulus in row (xii) is actually earned.
• Example 2 (Part II pp. 232–233), three parts, all of them rewriting before integrating. (i) A cube less one over a square, split into the variable and a negative power. (ii) The two-thirds power plus one. (iii) A three-halves power, plus twice an exponential, less the reciprocal of the variable. Verified, working added here: (i) gives half a square plus the reciprocal of the variable; (ii) gives three fifths of the five-thirds power plus the variable; (iii) gives two fifths of the five-halves power, plus twice the exponential, less the logarithm of the modulus. Each checked by differentiating. Note that part (i) also carries the chapter's Note about collapsing several constants into one — that Note belongs to the next topic and is cited there.
• Example 3 (Part II p. 233), three parts, all of them trigonometric rewrites. (i) A sine plus a cosine, done term by term. (ii) The cosecant times the sum of the cosecant and the cotangent, expanded into two rows of the list. (iii) One less a sine, all over the square of the cosine, split into the square of the secant and the secant times the tangent. Verified: (i) gives minus the cosine plus the sine; (ii) gives minus the cotangent less the cosecant; (iii) gives the tangent less the secant. Part (iii) is the one worth attention — no row of the list matches the integrand until the fraction is split, and after the split both pieces are rows (iv) and (vi).
• Exercise 7.1, questions 6 to 20 (Part II pp. 234–235). Fifteen items, all of them a rewrite followed by a table lookup. Worth flagging for the script: Q11, Q12 and Q13 divide a polynomial by a monomial or by a linear factor first; Q14, Q15 and Q19 need an index law or a ratio identity before any row applies; Q18 and Q20 are Example 3 (iii) in a different arrangement. Q13 is the only one that requires a division rather than a split, and a script that groups it with the others will mislead.
• Exercise 7.1, question 21 (Part II p. 235). Four options for the anti derivative of the sum of a square root and its reciprocal. Verified by working added here: the two terms are the half power and the minus-half power, so the answer is two thirds of the three-halves power plus twice the half power, which is the third option. The chapter prints no answers.
• The Summary's list (Part II pp. 287–288). Read cell by cell off the printed page images: fourteen numbered rows, (i) through (xiv). Thirteen of them correspond to the chapter's own list, with the last three reordered — the exponential, then the general base, then the reciprocal of the variable, where Part II pp. 228–229 runs exponential, reciprocal, general base. Row (xi) has no counterpart in the chapter's own list: it integrates the reciprocal of one plus a square and answers with minus the inverse cotangent. This is section 10, and the finding is set out in Notes below.

Figures to have open

• A two-column redraw of the Part II pp. 228–229 list for section 1, showing at least four rows in both directions. The content is the chapter's own; the arrows are added here. Build it with the repo's DataTable component.
• A six-row table of the trigonometric rows for section 3, with the three minus signs distinguished. Same component, same source pages.
• A side-by-side of the two lists for section 10 — thirteen rows against fourteen — with the unmatched row marked. The two lists are the chapter's own, on Part II pp. 228–229 and Part II pp. 287–288; the alignment is added here.
• A number line with zero excised, for section 5, to carry the two sign cases of Example 1 (iii). Not in the book; the chapter draws nothing.
• 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", the two-column list of standard formulae, Part II pp. 228–229
• The boxed Note on suppressed intervals, Part II p. 229
• The method-of-inspection paragraph and Example 1, §7.2.1, Part II pp. 231–232
• Examples 2 and 3, Part II pp. 232–233
• Exercise 7.1, questions 6 to 21, Part II pp. 234–235
• Summary, the standard-integral list, Part II pp. 287–288