Properties of the definite integral, and using symmetry to kill half the work

What to assume they know

• The second fundamental theorem, from the second topic of this module — every proof in this section runs through it
• Substitution with the limits moved, from the third topic of this module — five of the eight proofs are substitutions
• Even and odd functions, and the sign test on the negated input
• The trigonometric identities used to fold a squared ratio into a doubled angle
• The logarithm of a product, for the hardest example on these pages
• Sketching where a cubic changes sign, for the modulus example

What they should be able to do

• State each of the eight numbered properties and say what each one changes about an integral
• Identify which properties are essentially bookkeeping and which do real work
• Apply the reflection property to an integral that reappears after reflection, and solve for it by adding the integral to itself
• Recognise an even or odd integrand on a symmetric interval and write the answer down without integrating
• Split an interval at a sign change to remove a modulus
• Follow the chapter's longest example, including the step it marks with a question
• Choose a property from the shape of an integral rather than by trying them in turn
• State the general result that the last multiple-choice item of the chapter is a case of

Where it usually goes wrong

• "The properties are shortcuts for the exam." Three of them are needed to prove the others, and one exists for nothing else. Present the list as a structure with a shape, not as eight independent tricks.
• "Reflecting the interval gives you the answer." It gives you a second expression for the same number. The answer comes from what happens when you add the two — usually a collapse to something trivial. Teach the two moves as one move.
• "Even times odd, odd times odd — I will work the parity out on the spot." In Example 31 the parity of a product decides the whole question in one line, and it is the single most reliable mark in this section. Drill it separately.
• "A symmetric interval means the answer is zero." Only for an odd integrand. For an even one it means the answer is twice a half. Both halves of the parity property carry equal weight and students remember only the zero.
• "Bars around the integrand are a symmetry problem." They are a splitting problem. Find where the inside changes sign, cut there, and drop the bars with the right sign in each piece.
• "The reflection point is always the middle of a nice interval." In Example 33 it is the sum of a sixth and a third of a half turn, which is why the example is in the book at all. The property is stated for any pair of limits.
• "If reflecting does not simplify, the property was the wrong choice." In Example 34 reflecting produces something no simpler, and the simplification comes two steps later from a logarithm identity. Persistence is part of the method here.
• "The doubled-range property finishes the job." Exercise 7.10 Q14 shows it handing you a smaller integral that still needs work.

Questions to check understanding

• State a named property and say what it does to an integral
• Evaluate an integral that reappears under reflection, by adding it to itself
• Decide the parity of a product and write down a symmetric-interval integral without integrating
• Remove a modulus by splitting at the sign changes and evaluate — the form of Example 28 and of Exercise 7.10 Q5, Q6 and Q18
• Apply the doubled-range property and then finish with a second property — the form of Exercise 7.10 Q14
• Prove a stated identity from the reflection property — the form of Exercise 7.10 Q19
• Choose the correct value from four options where the odd parts vanish — the form of Exercise 7.10 Q20 and Q21
• State and prove the general reflection result for the variable times a function — the form of Miscellaneous Exercise Q40

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 list of eight properties (Part II pp. 273–274). Labelled from a zero-subscript first entry through a seventh, with the seventh split into two parts. Read off the page images. The whole list occupies just over one page, and the chapter's own reason for it is that the properties make definite integrals easier to evaluate. For section 1, show all eight at once and then dim all but the one under discussion — the group is small enough that a learner can hold it.
• The proofs (Part II pp. 274–276). Every one is short. The first is a bare renaming. The second and third go through the second fundamental theorem. The fourth, fifth, sixth and eighth are substitutions with the limits moved, which is why the previous topic is a hard prerequisite. The chapter proves all eight, which is worth saying aloud given that it proved neither fundamental theorem two pages earlier.
• The one property that never appears again (Part II p. 274, proved on Part II p. 275). The fifth splits an integral over a doubled range into two integrals over the single range, the second with the variable reflected. Established by working through every example and every exercise item in the section: it is used in exactly one place, the proof of the sixth property, and nowhere else — no example, no exercise, no miscellaneous item. It is a lemma wearing a property's number. Section 6 should say so, because a learner who tries to memorise all eight as equally useful is memorising one too many.
• Example 28 (Part II p. 276). The integral of the modulus of a cubic minus the variable, from minus one to two. The solution first states where the inside is positive and where negative, then splits at both sign changes and drops the bars with the right sign in each piece. Verified by working added here: the inside is non-negative on the leftmost stretch, non-positive on the middle one and non-negative on the rightmost; the three pieces come to a quarter, a quarter and nine quarters, and the total is eleven quarters, which is what the page prints. This is the only example in the section that uses no symmetry at all, and it is section 8. Note for the script: the splitting property as stated cuts an interval once, and this example cuts twice — a legitimate repeat, but say it rather than letting it pass.
• Example 29 (Part II pp. 276–277). A squared sine over a balanced interval. The solution names the integrand even and halves the work at once. Verified: the value is an eighth of a full turn less a half. Use it in section 7 as the shortest possible demonstration.
• Example 30 (Part II p. 277). The variable times a sine, over one plus a squared cosine, across a half turn. Three properties in one solution: reflect, get the same integral back with a constant times a simpler one, solve for it; then substitute for the cosine, flip the limits by the sign property, and finish with the even-function property. Verified: the value is the square of a half turn divided by four. This is the chapter's showcase for the add-it-to-itself move and belongs in section 5.
• Example 31 (Part II p. 277). A fifth power of sine times a fourth power of cosine, over a balanced interval. The solution tests the negated input, finds the whole product odd, and writes zero. Verified. One line. Pair it with Example 29: same reasoning, opposite parity, opposite outcome.
• Example 32 (Part II p. 278). A fourth power of sine over the sum of fourth powers of sine and cosine, across a quarter turn. Reflecting swaps sine and cosine, so adding the two forms collapses the fraction to one and the answer is half the length of the interval. Verified: a quarter of a half turn. The cleanest instance of the technique in the book — section 5 should run this one before Example 30, even though the chapter prints it after.
• Example 33 (Part II p. 278). One over one plus the square root of the tangent, between a sixth and a third of a half turn. The text layer of this page drops every radical in the example, turning the integrand into one over one plus the tangent and the rewriting into a ratio of plain cosines. The radicals are there. The two limits sum to a quarter turn, which is exactly what the reflection property needs, and adding the two forms gives twice the answer as the length of the interval. Verified: the value is a half turn divided by twelve. Use it in section 5 as the case where the reflection point is not the obvious midpoint of a standard interval.
• Example 34 (Part II p. 279). The logarithm of the sine across a quarter turn — the chapter's longest and hardest worked example, and section 9. It reflects to get the logarithm of the cosine, adds the two, inserts and removes a logarithm of two to turn a sum of logarithms into the logarithm of a doubled angle, marks that step with a printed question mark rather than justifying it, substitutes to restore the interval, and applies the sixth property. Verified: the value is minus a half turn over two, times the logarithm of two. The printed "Why?" is a gift for an explanation — stop there and answer it.
• Exercise 7.10, questions 1 to 21 (Part II p. 280). Grouped by property — an added classification, not the chapter's. Q1 is a plain identity; Q2, Q3, Q4, Q7, Q8, Q12, Q15, Q16, Q17 and Q19 are reflections, most of them the add-it-to-itself move; Q5, Q6, Q9 and Q18 involve a modulus or a root and are split or reflected without symmetry cancelling anything; Q10 leans on the result of Example 34; Q11, Q13 and Q20 are parity; Q14 needs the doubled-range property and then a second step. Q21 is parity in disguise. Verified in full: Q13 and Q15 are zero by odd parity and reflection respectively; Q17 is half the upper limit; Q19's stated identity follows in two lines from the reflection property; Q20 is a half turn and Q21 is zero, each matching a printed option.
• Exercise 7.10, question 14 (Part II p. 280). A fifth power of cosine across a full turn. Verified: the doubled-range property applies and gives twice the integral across a half turn, and that is not yet zero — a second reflection is needed to finish. It is the only item in the exercise that takes two properties in sequence, and it is worth one slide of its own inside section 10.
• Miscellaneous Exercise, question 40 (Part II p. 287). Given that a function is unchanged when its input is reflected through the sum of the two limits, the integral of the variable times that function is asked for. Verified by working added here: reflecting and adding gives the sum of the limits, halved, times the integral of the function alone — the fourth option. This is Example 30 with the particular function stripped out, and section 11 should show that: the example is the general rule at one specific pair of limits. Note the distractors are built well — two carry the right factor with a mangled integrand, and one carries the right integrand with the difference of the limits instead of the sum — so a half-remembered rule lands on a wrong option rather than on nothing.
• This topic uses no figure from the chapter. The chapter's only figure, on Part II p. 267, belongs to g12-maths-ch07-m03-t02.md.

Figures to have open

• A sign chart for section 8: the cubic minus the variable plotted across the whole range with the two crossings marked and the three pieces shaded differently. Not in the book; the chapter states the signs in a sentence and draws nothing.
• A dependency graph of the eight properties for section 6, edges drawn from the chapter's own proofs on Part II pp. 274–276, so that the lemma-only property is visibly a dead end. Build it with the repo's Network component. The graph is added here; every edge is read off a printed proof.
• Two reflection diagrams for section 4: an interval with an arrow folding it onto itself, once about the midpoint of a general pair of limits and once about the midpoint of an interval starting at zero. Not in the book.
• A grouped table of Exercise 7.10 for section 10, and a second small table for section 1 holding all eight properties. The items and the properties are the chapter's own, on Part II pp. 273–274 and Part II p. 280; the grouping and the glosses are added here. Both with the repo's DataTable component.
• A side-by-side of Example 30 and the last miscellaneous item for section 11, aligned line for line so the general rule and its instance sit on the same rows. Not in the book.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 7 "Integrals", §7.10 Some Properties of Definite Integrals, the list of eight, Part II pp. 273–274
• The eight proofs, Part II pp. 274–276
• Examples 28 and 29, Part II pp. 276–277
• Examples 30 and 31, Part II p. 277
• Examples 32 and 33, Part II p. 278
• Example 34, Part II p. 279
• Exercise 7.10, questions 1 to 21, Part II p. 280
• Miscellaneous Exercise on Chapter 7, question 40, Part II p. 287