alleryThe 'King's property' of definite integration is:
A$\int_a^b f(x)\,dx = \int_a^b f(b-x)\,dx$
B$\int_a^b f(x)\,dx = \int_a^b f(a+b-x)\,dx$
C$\int_a^b f(x)\,dx = \int_a^b f(x+a)\,dx$
D$\int_a^b f(x)\,dx = \int_a^b f(b/x)\,dx$
Answer & Solution
Correct answer: B. $\int_a^b f(x)\,dx = \int_a^b f(a+b-x)\,dx$
**Property V:** substitute $t = a + b - x$.
The interval $[a, b]$ maps to itself reversed, giving $\int_a^b f(x)\,dx = \int_a^b f(a+b-x)\,dx$.
Useful for symmetric simplification.
Keep practising Unit 6B: Definite Integration & Riemann Sums
Browse all Unit 6B: Definite Integration & Riemann Sums questions for AP Calculus AB, or explore more Mathematics topics.
Related questions
∫ from 0 to π/2 sin x cos x dx =For ∫ dx / sqrt(a² - x²), let x = a sin θ. Result:∫ from 0 to 2 (x² - 2x) dx =Reduction formula example: ∫ sinⁿ x dx (for n ≥ 2) =∫ from 1 to e (1/x) dx =∫ x ln x dx (integration by parts: u = ln x, dv = x dx):∫ from 0 to ∞ e^(-x) dx =Area under y = x² from x = 0 to x = 2: