Definite Integrals: Fundamental Theorem & Properties

• State both parts of the Fundamental Theorem of Calculus and use them
• Apply the seven standard properties of definite integrals
• Use the King's property to evaluate integrals that are otherwise hard
• Evaluate integrals of even and odd functions over symmetric limits efficiently
• Recognize when to apply the splitting and substitution properties

Level 1 — Foundations

A definite integral ∫ₐᵇ f(x)dx represents the signed area between the curve y=f(x) and the x-axis from x=a to x=b.

Unlike an indefinite integral, the result is a number, not a function.

Fundamental Theorem of Calculus (FTC), Part 2: If F is an antiderivative of f on [a,b], then: $\int_a^b f(x)\,dx = F(b) - F(a) = \Big[F(x)\Big]_a^b$

This is how we evaluate definite integrals in practice — find the antiderivative, then substitute limits.

Level 2 — JEE depth

FTC Part 1 (Leibniz Rule): $\frac{d}{dx}\left[\int_a^x f(t)\,dt\right] = f(x)$

Extension (chain rule form): $\frac{d}{dx}\left[\int_{g(x)}^{h(x)} f(t)\,dt\right] = f(h(x))\cdot h'(x) - f(g(x))\cdot g'(x)$

Properties of Definite Integrals:

P1 (Reversal): $\displaystyle\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx$

P2 (Zero width): $\displaystyle\int_a^a f(x)\,dx = 0$

P3 (Additivity): $\displaystyle\int_a^b f(x)\,dx + \int_b^c f(x)\,dx = \int_a^c f(x)\,dx$

P4 (Dummy variable): $\displaystyle\int_a^b f(x)\,dx = \int_a^b f(t)\,dt$

P5 (Translation): $\displaystyle\int_a^b f(x)\,dx = \int_a^b f(a+b-x)\,dx$ ← King's Property

P6 (Half-period): $\int_0^{2a} f(x)\,dx = \begin{cases} 2\int_0^a f(x)\,dx & \text{if } f(2a-x) = f(x) \\ 0 & \text{if } f(2a-x) = -f(x) \end{cases}$

P7 (Even/Odd over symmetric limits): $\int_{-a}^{a} f(x)\,dx = \begin{cases} 2\int_0^a f(x)\,dx & \text{if } f \text{ is even } [f(-x)=f(x)] \\ 0 & \text{if } f \text{ is odd } [f(-x)=-f(x)] \end{cases}$

King's Property in use: Add the original integral I to I with King's substitution to find 2I, then divide. Classic template: $I = \int_0^{\pi} \frac{x\,\sin x}{1+\cos^2 x}\,dx \xrightarrow{\text{King's}} I = \int_0^{\pi} \frac{(\pi-x)\sin x}{1+\cos^2 x}\,dx$ $\Rightarrow 2I = \pi\int_0^{\pi} \frac{\sin x}{1+\cos^2 x}\,dx$

Evaluate I = ∫₀^(π/2) ln(sin x) dx

Step 1: Apply King's property with a=π/2:
  f(π/2 − x) = ln(sin(π/2 − x)) = ln(cos x)
  So I = ∫₀^(π/2) ln(cos x) dx   (same value by King's)

Step 2: Add original and King's version:
  2I = ∫₀^(π/2) [ln(sin x) + ln(cos x)] dx
     = ∫₀^(π/2) ln(sin x · cos x) dx
     = ∫₀^(π/2) ln(sin 2x / 2) dx
     = ∫₀^(π/2) ln(sin 2x) dx − ∫₀^(π/2) ln 2 dx

Step 3: Substitute u = 2x in first integral:
  ∫₀^(π/2) ln(sin 2x) dx = (1/2)∫₀^π ln(sin u) du
  By half-period property (sin(π−u) = sin u, so f is symmetric):
  = (1/2) · 2 · ∫₀^(π/2) ln(sin u) du = I

Step 4: 2I = I − (π/2)ln 2
  I = −(π/2)ln 2

Answer: ∫₀^(π/2) ln(sin x) dx = −(π ln 2)/2

• Evaluate ∫₀^π x/(1+sin x) dx using King's property
• Show that ∫₋₂² x³ cos x dx = 0 (odd function)
• Evaluate ∫₋₁¹ (x² + |x| + 1)/(x²+1) dx (split into even + odd parts)
• Use FTC Part 1 to find d/dx[∫₁^(x²) sin t dt]
• Evaluate ∫₀^(π/2) (sin x)/(sin x + cos x) dx

• Fig 7.1: Area as the limit of Riemann sums — geometric motivation for FTC
• The standard result ∫₀^(π/2) sin x dx = ∫₀^(π/2) cos x dx = 1 (follows from King's)

• ∫₀^π sin x dx = ? → 2
• Is x⁵ + x³ even or odd? → Odd; so ∫₋₁¹ (x⁵+x³)dx = 0
• ∫ₐᵇ f(x)dx + ∫ᵦᵃ f(x)dx = ? → 0
• FTC Part 1: d/dx[∫₃^x t² dt] = ? → x²
• Stretch: Prove that ∫₀¹ xⁿ(1−x)ⁿ dx = ∫₀¹ xⁿ(1−x)ⁿ dx using the substitution x→1−x, and comment on what this tells you.

NCERT Chapter 7 link: Integrals — definite integrals start at section 7.7; properties in section 7.10 with solved examples 35–40 and Exercise 7.11
Exam connections: Properties are used heavily in JEE problems involving log-trig integrals, inverse trig integrals, and Gamma function style problems; also links to area (Chapter 8)
Study strategy: King's property solves roughly 30% of JEE definite integral questions. Drill 10 King's problems; then practice even/odd problems. Only after that move to area applications.

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