Definite Integration

🟢 Lite — Quick Review

Rapid summary for last-minute revision before your exam.

Definite Integration — Key Facts for JEE Main Definite integral: ∫_a^b f(x) dx = F(b) − F(a) where F is antiderivative of f Fundamental theorem of calculus: d/dx [∫_a^x f(t) dt] = f(x) Properties: ∫_a^b f(x) dx = −∫_b^a f(x) dx; ∫_a^b f(x) dx = ∫_a^c f(x) dx + ∫c^b f(x) dx Even function: ∫{−a}^a f(x) dx = 2∫0^a f(x) dx; Odd function: ∫{−a}^a f(x) dx = 0 ⚡ Exam tip: For periodic functions or symmetric limits, use properties to simplify before evaluating!

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🟡 Standard — Core Study

Standard content for students with a few days to months.

Definite Integration — JEE Main Study Guide

Fundamental Theorem of Calculus: If F’(x) = f(x), then ∫_a^b f(x) dx = F(b) − F(a) Also: d/dx [∫_a^x f(t) dt] = f(x); this is the first FTC

Evaluation tricks:

• Change variable and limits simultaneously in substitution
• For definite integral of rational function with denominator that factors: use partial fractions

Symmetry properties:

• f even (f(−x) = f(x)): ∫_{−a}^a f(x) dx = 2∫_0^a f(x) dx
• f odd (f(−x) = −f(x)): ∫_{−a}^a f(x) dx = 0

Periodic function: If f(x + T) = f(x), then ∫_a^{a+T} f(x) dx is independent of a

King’s property: ∫_a^b f(x) dx = ∫_a^b f(a + b − x) dx Useful for integrals where f has symmetry about midpoint (a+b)/2

Integral as limit of sum: ∫_a^b f(x) dx = lim(n→∞) Σ f(a + k·h)·h where h = (b−a)/n, k = 0 to n−1

Gamma and Beta: Γ(n) = (n−1)! for positive integer n; Γ(1/2) = √π B(m, n) = 2∫_0^{π/2} sin^{2m−1}x cos^{2n−1}x dx = Γ(m)Γ(n)/Γ(m+n)

Walli’s integrals: ∫_0^{π/2} sin^m x dx = (m−1)!!/(m)!! × π/2 if m is even; = (m−1)!!/(m)!! if m is odd ∫_0^{π/2} sin^m x cos^n x dx = use Beta function

Walls formula: ∫_0^{π/2} sin^m x dx = ∫_0^{π/2} cos^m x dx

• Key formula: ∫a^b f(x) dx = F(b) − F(a); ∫{−a}^a f(x) dx = 2∫_0^a f(x) dx for even, 0 for odd
• Common trap: Don’t forget to adjust limits when doing substitution in definite integrals
• Exam weight: 1–2 questions per year (4–8 marks); highly scoring with proper technique

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🔴 Extended — Deep Dive

Comprehensive coverage for students on a longer study timeline.

Definite Integration — Comprehensive JEE Main Notes

Leibniz rule for differentiation under integral: d/dx [∫{φ(x)}^{ψ(x)} f(t) dt] = f(ψ(x))·ψ’(x) − f(φ(x))·φ’(x) Extension: d/dx [∫{a}^{b} f(x,t) dt] = ∫_{a}^{b} ∂f/∂x (x,t) dt

Evaluation using series: ∫_0^1 x^n dx = 1/(n+1); ∫_0^1 x^{p−1}(1−x)^{q−1} dx = B(p,q) = Γ(p)Γ(q)/Γ(p+q)

Integral of greatest integer function: ∫0^n [x] dx = Σ{k=0}^{n−1} k = n(n−1)/2

Trapezoidal rule approximation: ∫a^b f(x) dx ≈ (h/2)[f(x_0) + 2f(x_1) + 2f(x_2) + … + 2f(x{n−1}) + f(x_n)] Where h = (b−a)/n and x_i = a + ih

Simpson’s rule (1/3 rule): ∫a^b f(x) dx ≈ (h/3)[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + … + 4f(x{n−1}) + f(x_n)] For even number of intervals n

Area between curves: Area = ∫_a^b |f(x) − g(x)| dx For parametric (x(t), y(t)): area = ∫ y(t)·x’(t) dt

Area with polar coordinates: Area = (1/2)∫ r² dθ between appropriate θ values

Volume by slicing: V = ∫ A(x) dx where A(x) is cross-sectional area perpendicular to x-axis

Reduction formulas for definite integrals: I_n = ∫0^{π/2} sin^n x dx = (n−1)/n · I{n−2} I_n = (n−1)/n · (n−3)/(n−2) · … · π/2 (for even n) or 1 (for odd n)

Special definite integrals:

• ∫_0^∞ e^{−ax} dx = 1/a
• ∫_0^∞ x e^{−ax} dx = 1/a²
• ∫_0^∞ x² e^{−ax} dx = 2/a³
• ∫_0^∞ sin x/x dx = π/2
• ∫_0^∞ cos x/x dx = divergent

Convergence of improper integrals: ∫a^∞ f(x) dx converges if lim{b→∞} ∫_a^b f(x) dx exists For ∫_a^∞ e^{−kx} dx: converges if k > 0, diverges if k ≤ 0

Dirichlet’s test: If f is bounded and monotonically decreasing to 0, and g has bounded integral, then ∫_a^∞ f(x)g(x) dx converges

Using king’s property: I = ∫_a^b f(x) dx I = ∫_a^b f(a+b−x) dx (by substitution u = a+b−x) Adding: 2I = ∫_a^b [f(x) + f(a+b−x)] dx This is useful when f(x) + f(a+b−x) simplifies

Application: proving integrals: Often asked to prove value of definite integral using symmetry Example: I = ∫_0^{π/2} ln(sin x) dx = −(π/2) ln 2

• Remember: ∫_a^b f(x) dx = F(b) − F(a); if f(−x) = f(x) → even, double from 0; if f(−x) = −f(x) → odd, zero; king’s property: ∫_a^b f(x) dx = ∫_a^b f(a+b−x) dx
• Previous years: “Evaluate ∫_0^π x sin x/(1 + cos²x) dx” [2023]; “Show that ∫_0^a f(x) dx + ∫_0^a f(a−x) dx = a∫_0^a f(x) dx” [2024]; “Evaluate ∫_0^{π/2} sin²x dx” [2024]

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📊 JEE Main Exam Essentials

Detail	Value
Questions	90 (30 per subject)
Time	3 hours
Marks	300 (90 per subject)
Section	Physics (30), Chemistry (30), Mathematics (30)
Negative	−1 for wrong answer
Mode	Computer-based

🎯 High-Yield Topics for JEE Main Mathematics

• Calculus (Differentiation + Integration) — ~35 marks combined
• Coordinate Geometry (straight lines, circles, conics) — ~20 marks
• Algebra (Complex Numbers, Quadratics, P&C, Probability) — ~25 marks
• Trigonometry + Inverse Trigonometry — ~15 marks
• Vector + 3D — ~15 marks

📝 Previous Year Question Patterns

• Definite Integration: 1–2 questions per year, 4–8 marks
• Common patterns: evaluate using symmetry, using king’s property, area under curve, Wallis formula
• Weight: medium-high frequency, high scoring

💡 Pro Tips

• Always check for symmetry before doing full calculation — it can save significant time
• For integrals with periodic functions, use the periodicity property
• King’s property (a+b−x) is very powerful for integrals that aren’t symmetric but have midpoint symmetry
• When evaluating improper integrals, check convergence first
• The limit-of-sum definition of definite integral can be used to evaluate tricky sums
• For Wallis-type integrals, know the reduction formulas

🔗 Official Resources

• NTA Official JEE Main
• JEE Main Syllabus PDF

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