Integration

🟢 Lite — Quick Review

Rapid summary for last-minute revision before your exam.

Integration — Key Facts for JEE Main Integral: reverse of differentiation; ∫ f(x) dx = F(x) + C where F’(x) = f(x) Indefinite integral: ∫ f(x) dx = F(x) + C (family of curves) Definite integral: ∫_a^b f(x) dx = F(b) − F(a) (area under curve from a to b) Standard integrals: ∫ x^n dx = x^{n+1}/(n+1) + C (n ≠ −1); ∫ 1/x dx = ln|x| + C; ∫ e^x dx = e^x + C ⚡ Exam tip: For definite integrals, always check if the integrand is symmetric — odd functions integrate to 0 over symmetric limits!

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🟡 Standard — Core Study

Standard content for students with a few days to months.

Integration — JEE Main Study Guide

Basic integration formulas:

• ∫ x^n dx = x^{n+1}/(n+1) + C (n ≠ −1)
• ∫ 1/x dx = ln|x| + C
• ∫ e^x dx = e^x + C
• ∫ a^x dx = a^x/ln a + C
• ∫ sin x dx = −cos x + C
• ∫ cos x dx = sin x + C
• ∫ sec²x dx = tan x + C
• ∫ cosec²x dx = −cot x + C

Integration methods:

Substitution: Let u = g(x), then du = g’(x)dx Transform integral ∫ f(g(x))·g’(x) dx into ∫ f(u) du For definite integrals: change limits to u-values

Parts formula: ∫ u dv = uv − ∫ v du Choose u using LIATE: Log, Inverse trig, Algebraic, Trigonometric, Exponential For integrals like x·e^x, x·sin x, x·ln x → use parts

Partial fractions: For rational functions, decompose into partial fractions Linear factors: A/(x−a), Bx+C/(x²+bx+c) After decomposition, integrate each term separately

Standard substitutions:

• For √(a² − x²): x = a sin θ or a cos θ
• For √(a² + x²): x = a tan θ
• For √(x² − a²): x = a sec θ

Properties of definite integrals:

• ∫_a^b f(x) dx = −∫_b^a f(x) dx
• ∫_a^b f(x) dx + ∫_b^c f(x) dx = ∫_a^c f(x) dx
• Even function: ∫_{−a}^a f(x) dx = 2∫_0^a f(x) dx
• Odd function: ∫_{−a}^a f(x) dx = 0

Leibniz rule: d/dx [∫_{a}^{b} f(x,t) dt] = f(x,b)·db/dx − f(x,a)·da/dx + ∫_a^b ∂f/∂x dt

• Key formula: ∫ x^n dx = x^{n+1}/(n+1) + C; ∫ 1/x dx = ln|x| + C; ∫ e^x dx = e^x + C
• Common trap: For definite integrals with symmetric limits on odd/even functions, use the properties — don’t compute the full integral
• Exam weight: 2–3 questions per year (8–12 marks); high-scoring if techniques are mastered

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🔴 Extended — Deep Dive

Comprehensive coverage for students on a longer study timeline.

Integration — Comprehensive JEE Main Notes

Integration of trigonometric functions:

• ∫ sin^n x dx: use power reduction formulas
• ∫ cos^n x dx: same approach
• ∫ sin x cos x dx = (1/2)∫ sin 2x dx = −(1/4) cos 2x + C
• ∫ tan x dx = −ln|cos x| + C = ln|sec x| + C

Using half-angle identities: sin²x = (1−cos 2x)/2; cos²x = (1+cos 2x)/2 ∫ sin²x dx = x/2 − sin 2x/4 + C

Reduction formulas: For ∫ sin^n x dx, use: I_n = −(sin^{n−1}x cos x)/n + (n−1)/n · I_{n−2} Similarly for cos^n x and other powers

Integration of inverse trigonometric:

• ∫ sin⁻¹x dx = x sin⁻¹x + √(1−x²) + C
• ∫ cos⁻¹x dx = x cos⁻¹x − √(1−x²) + C
• ∫ tan⁻¹x dx = x tan⁻¹x − (1/2) ln(1+x²) + C

Integration of exponential times polynomial: ∫ x^n e^x dx: use parts repeatedly, taking polynomial as u each time I_n = x^n e^x − n I_{n−1}

Walli’s formula for definite integrals: ∫_0^{π/2} sin^m x cos^n x dx = (Γ((m+1)/2)·Γ((n+1)/2)) / (2·Γ((m+n+2)/2))

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

Area under curve: Area between y = f(x) and x-axis from x=a to x=b = ∫_a^b |f(x)| dx Area between two curves: ∫_a^b |f(x) − g(x)| dx For parametric (x(t), y(t)): area = ∫ y(t) · x’(t) dt

Volume of revolution: About x-axis: V = π∫ y² dx; About y-axis: V = π∫ x² dy Using cylindrical shells: V = 2π∫ x·f(x) dx (about x-axis, using vertical strips)

Using properties for even/odd definite integrals:

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

Integral with periodicity: If f(x+T) = f(x), then ∫_a^{a+T} f(x) dx is same for any starting point a

Special tricks: ∫_0^∞ x^n e^{−x} dx = n! ∫_0^{π/2} ln(sin x) dx = −(π/2) ln 2

Differential equation via integration: dy/dx = f(x) → dy = f(x)dx → ∫ dy = ∫ f(x)dx → y = ∫ f(x)dx + C

Average value of function: Average = (1/(b−a)) ∫_a^b f(x) dx

• Remember: ∫ f’(x) dx = f(x) + C (reverse of differentiation); for even functions over symmetric limits, double from 0 to a; for odd functions over symmetric limits, result is 0; use parts when integrand is product of polynomial and exponential/trig function
• Previous years: “Evaluate ∫_0^1 x e^x dx using integration by parts” [2023]; “Find area bounded by y = x² and y = x” [2024]; “Evaluate ∫ sin⁻¹x 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

• Integration: 2–3 questions per year, 8–12 marks
• Common patterns: evaluate definite integrals, find area under curve, integration by parts, trigonometric integrals
• Weight: very high frequency, requires extensive practice

💡 Pro Tips

• For integration by parts: choose u using LIATE (Log, Inverse, Algebraic, Trigonometric, Exponential)
• When integrating rational functions, always use partial fractions if degree of numerator < degree of denominator
• For symmetric definite integrals, check if f(x) is even or odd before computing
• Standard substitutions are crucial: √(a²−x²) → x = a sin θ; √(a²+x²) → x = a tan θ; √(x²−a²) → x = a sec θ
• For definite integrals with infinity limits, split into proper limits and evaluate
• In area problems, sketch the curves first to identify which curve is on top in the interval

🔗 Official Resources

• NTA Official JEE Main
• JEE Main Syllabus PDF

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