Math notesCalculusIntegrals
Independent topic page
Integrals full marks: property glance (King/even-odd) before expanding → technique if needed → for area, intersections first then ∫|f−g|.
01. King’s rule: ∫_a^b f(x)dx = ∫_a^b f(a+b−x)dx. Classic: ∫₀ᵃ f/(f(x)+f(a−x)) = a/2 when defined.
02. Even/odd about 0: ∫_{-a}^{a} even = 2∫₀ᵃ; odd = 0 (when exists).
03. Area is geometry+calculus: solve f=g for bounds, test which curve is above, integrate absolute difference on each piece.
04. FTC: d/dx ∫_a^{g(x)} f(t)dt = f(g(x))g′(x) (standard Leibniz form used in Advanced).
05. Advanced lane: definite-integral properties (even/odd, King property, constant limits tricks) beat expansion. Look for f(x)+f(a−x) structure before any antiderivative hunt.
06. Area between curves: find intersections first and fix limits; wrong bounds waste the entire attempt even if the integrand is perfect.
07. Rational integrands: decide partial fractions vs substitution early; ILATE for parts when the product is polynomial×log/exp/trig.
Official · 2026
Official syllabus points for Integrals · Properties · Area only — not the full Calculus portion map. Portion hub: Calculus · Calculus analysis.
No topic-specific exclusions flagged — finish the unit points below for Integrals.
Source: https://jeeadv.ac.in/documents/jee-advanced-2026-syllabus.pdf
Exam focusIndefinite/definite integrals and area under curves — not DE methods.
Syllabus points
Do / check
2020–2026 paper weight
Topic match inside parent portion Calculus (28.5% of Math tags). Stats below are portion backdrop; matched subtopics show where Integrals · Properties · Area sits.
Portion-wide type/difficulty mix lives on the Calculus analysis — not repeated here so this page stays on Integrals.
01. Dominant subtopics: AOD (18), Definite Integrals (11), Continuity/Differentiability (9), Functions (7).
Cross-topic stems
Advanced rarely asks a chapter in isolation. These are tagged multi-idea Math stems that touch Integrals · Properties · Area — practise the knowledge bridge, not only the single-chapter drill.
Step 1: set y = e^{-x} equal to y = e^{-x}(sin x + cos x) and cancel the common exponential to get a trig equation for intersection abscissae α_i. Step 2: order those roots and set up the definite integral of the difference of the two curves between α_1 and α_4 for the enclosed area β. Step 3: simplify the given log expression in β to a clean numerical value.
The stem hides trig solving inside an exponential envelope; students often integrate before finding exact limits.
Drill ‘cancel common positive factor → trig roots → area integral’ as one reflex chain.
Step 1: solve the two ellipse equations simultaneously for the first-quadrant intersection P. Step 2: differentiate implicitly to get slopes of both tangents at P; convert the angle formula into 4 tan θ. Step 3: for the common area, set up the integral (or standard ellipse-sector form) of the overlapping region and read cot α from that area.
Two different asks on one geometry figure — angle then area — under numerical fatigue.
For two conics, always compute intersection → slopes → area in that fixed order.
Step 1: analyse the given limit expression to identify the correct f(x) / derivative behaviour. Step 2: on a related (or following) stem, sketch the region defined by inequalities. Step 3: integrate carefully (logs appear from 1/x-type boundaries).
Back-to-back calculus depth: analytic limit then geometric area.
Pair every limit/AOD drill with one area-of-region numerical the same day.
Step 1: for the ellipse–parabola common-tangent MC, write both tangent conditions and compare. Step 2: later area NUM — find intersection limits first. Step 3: integrate difference of functions; keep trig-equation NUM technique nearby for neighbouring items.
Classic Advanced join: geometry conditions feeding calculus numerics in the same paper.
Any week you revise conics, end with one area-between-curves NUM.
Step 1: for the area item, find intersection limits and integrate. Step 2: for the parabola item, apply focus–directrix or tangent properties. Step 3: notice how often Advanced seats Calculus and Conics as neighbours — train that adjacency.
2020’s Calculus weight makes area items feel easy until limits are wrong.
Always find limits from geometry before writing ∫.
Why methods matter
Definite integrals in Advanced are often property contests. These methods put King/intersections before antiderivatives — that is how ugly integrals finish in under a minute.
Ugly definite integrals become constants — this is the Advanced definite-integral mindset.
Do thisSpend 10s on King/even-odd before antiderivative struggle.
Many Advanced integrals collapse to constants.
∫₀^{π/2} sin/(sin+cos) → King → I=π/4.
Wrong limits waste the whole attempt; this method makes bounds non-negotiable.
Do thisSolve f=g; order roots; test above/below.
Wrong limits are the #1 area killer.
y=x² and y=x meet at 0,1; area ∫₀¹(x−x²)=1/6.
Stops thrashing between parts and substitution under pressure.
Do thisChoose u via Inverse/Log/Algebraic/Trig/Exp preference.
Standardises parts selection.
∫ x e^x : u=x, dv=e^x dx.
Rationals get a clear recipe instead of random substitutions.
Do thisFactor denominator; decompose before integrating rational functions.
Avoids wrong substitutions on proper rationals.
Linear factors → A/(x−r) terms; irreducible quadratic → linear over quadratic.
Processed examples
Read the impact line first. Then notice which family/model was chosen. The solution is written the way your rough sheet should look in the hall.
Evaluate I = ∫₀^{π/2} sin x / (sin x + cos x) dx.
This is the poster child for King property — antiderivative struggle is optional.
Model chosenx → π/2 − x; add to get 2I = π/2.
Answerπ/4
Takeaway: Property before expansion on [0,a] uglies.
Area between y=x and y=x² from their intersection points.
Intersections first — wrong limits are the #1 area killer.
Model chosenSolve x=x² → bounds 0,1; integrate (x−x²).
Answer1/6
Takeaway: Write intersections before integrating.
Find the exact area bounded by y = |x² − 1|, the x-axis, and the lines x = −2 and x = 2.
Zeros of the expression force split limits — absolute-value areas without splits are wrong.
Model chosenEven ⇒ 2 × area on [0, 2]; split at x = 1.
Answer4
Takeaway: Mark every zero/crossing before integrating an absolute-value area.
Reusable models
Advanced stems cluster. Name the family in 10–15s, write the fast model, then run the same path. Do not invent a new plan each time.
Looks likeUgly integrand on [0,a] or [a,b] with hidden symmetry
Fast modelI=∫f; substitute x→a+b−x (or a−x); add; solve I — before antiderivative struggle.
Many Advanced integrals collapse to a constant in 30–45s.
Do not hunt a hard antiderivative before trying King.
Looks likeBounded area; curve and axis; two curves
Fast modelSolve intersections FIRST; test which is above; ∫|f−g| piecewise.
Wrong limits waste the whole attempt — intersections first prevents that.
Do not guess limits from a rough sketch alone.
Looks likeIndefinite integral of product / rational / composite
Fast model10s classify: sub if inner derivative visible; parts if product; PF if proper rational.
Classification prevents thrashing between methods.
Do not start parts when a substitution is obvious.
Looks likeDefinite integral symmetric about 0
Fast modelOdd → 0 (if exists); even → 2∫₀ᵃ — check continuity at 0.
Instant evaluation without antiderivative.
Do not integrate a clearly odd continuous function from −a to a the long way.
Hall timing
If limits 0 to a or −a to a, property is likely.
For area: solve intersections immediately; do not guess limits from a rough figure alone.
Keep constants exact. Use |f−g| when needed.
MC: verify each integral claim independently.
After the model is locked
∫_a^b f(x)dx = ∫_a^b f(a+b−x)dx
Symmetric definite limits
∫₀ᵃ f/(f(x)+f(a−x)) dx = a/2
Standard Advanced collapse
∫_{-a}^{a} even = 2∫₀ᵃ; odd = 0
Symmetric about 0
∫_a^b f = F(b)−F(a)
Antiderivative known
∫ u dv = uv − ∫ v du
Product integrands (ILATE pick u)
∫_a^b |f−g| dx on proper bounds
Area between curves
∫_a^b f = −∫_b^a f
Limit order fixes
t=tan(x/2) / x=a sinθ / etc. when structure matches
Indefinite technique
Make it automatic