Integral Calculus (Definite Integrals & Area)
Practice JEE Main Definite Integrals & Area Under Curves PYQs online. Master King’s Property, Leibniz Rule, periodic integrals & 4-step solutions.
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4-Step Solution & Tricks
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🎯 High-Yield Syllabus Breakdown
These high-frequency concepts account for over 90% of all questions asked in Integral Calculus (Definite Integrals & Area):
- King’s Property: ∫_a^b f(x) dx = ∫_a^b f(a + b - x) dx
- Leibniz Integral Rule for Differentiating under Integral Sign
- Even and Odd Functions Definite Integrals
- Periodic Functions & Definite Integrals (∫_0^nT f(x) dx = n ∫_0^T f(x) dx)
- Area Bounded Between Curves & Coordinate Axes
⚡ Essential Formulas & Speed Shortcuts
Formulas and time-saving shortcuts to solve Definite Integrals & Area questions in under 45 seconds:
King’s Property shortcut: If f(x) + f(a + b - x) = c (constant), then ∫_a^b f(x) dx = c · (b - a) / 2.
Leibniz Rule: d/dx [∫_u(x)^v(x) f(t) dt] = f(v(x))·v’(x) - f(u(x))·u’(x).
Area between parabola y² = 4ax and line y = mx is 8a² / (3m³).
📖 Sample Solved Exam Questions
Preview authentic questions from this chapter with full step-by-step solutions:
💡 View Step-by-Step Solution & Answer
Correct Answer: π/4
- Step 1: Let I = ∫₀^(π/2) [sin³ x / (sin³ x + cos³ x)] dx.
- Step 2: Apply King's Property: ∫₀^a f(x) dx = ∫₀^a f(a - x) dx. Here a = π/2.
- Step 3: Replace x with π/2 - x: I = ∫₀^(π/2) [cos³ x / (cos³ x + sin³ x)] dx.
- Step 4: Add the two expressions for I: 2I = ∫₀^(π/2) [(sin³ x + cos³ x)/(sin³ x + cos³ x)] dx = ∫₀^(π/2) 1 dx = π/2. Thus I = π/4.
⚡ Speed Tip: King's Property Symmetry: For ∫₀^(π/2) f(sin x)/[f(sin x) + f(cos x)] dx, the integral is always (b - a)/2 = π/4.
💡 View Step-by-Step Solution & Answer
Correct Answer: 4/3
- Step 1: Find intersection points: x² = 2x ⇒ x(x - 2) = 0 ⇒ x = 0 and x = 2.
- Step 2: On the interval [0, 2], the line y = 2x lies above the parabola y = x².
- Step 3: Area = ∫₀² (2x - x²) dx = [x² - x³/3]₀².
- Step 4: Evaluate at limits: (2² - 2³/3) - 0 = (4 - 8/3) = (12 - 8)/3 = 4/3.
⚡ Speed Tip: Archimedes Area Formula: Area between parabola y² = 4ax and line y = mx is 8a²/(3m³). For x² = 2x, [x² - x³/3]₀² = 4/3.
💡 View Step-by-Step Solution & Answer
Correct Answer: π/2
- Step 1: Apply Leibniz Rule for differentiation under the integral sign: d/dx ∫_{u(x)}^{v(x)} g(t) dt = g(v(x)) · v'(x) - g(u(x)) · u'(x).
- Step 2: Here upper limit v(x) = x with v'(x) = 1, and lower limit is constant 0 with u'(x) = 0.
- Step 3: Therefore, f'(x) = (x sin x) · (1) - 0 = x sin x.
- Step 4: Evaluate at x = π/2: f'(π/2) = (π/2) · sin(π/2) = (π/2) · 1 = π/2.
⚡ Speed Tip: Leibniz Rule: d/dx ∫₀ˣ g(t) dt = g(x). Simply evaluate x sin x at π/2 to get π/2.
❓ Frequently Asked Questions (FAQs)
How does King’s Rule simplify trigonometric integrals?
Adding the original integral I to King’s transformed integral gives 2I = ∫ (numerator / denominator) dx = ∫ 1 dx = (b - a), yielding I = (b - a)/2 immediately.