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PUCET (Punjab) King's Property — practice questions

Explore 4 free King's Property questions in PUCET (Punjab) Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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King's Property questions and exercises with answers

Showing questions 1–4 of 4.

The 'King's property' of definite integration is:Read the answer and worked solution →Using the King's property, $\int_0^a f(x)\,dx$ equals:Read the answer and worked solution →Using the King's property, evaluate $\int_0^{\pi/2} \dfrac{dx}{1 + \sqrt[3]{\tan x}}$:Read the answer and worked solution →$\int_0^{\pi/2} \cos^2 x\,dx$ equals:Read the answer and worked solution →

Worked practice questions

The 'King's property' of definite integration is:

  1. $\int_a^b f(x)\,dx = \int_a^b f(b-x)\,dx$
  2. $\int_a^b f(x)\,dx = \int_a^b f(a+b-x)\,dx$
  3. $\int_a^b f(x)\,dx = \int_a^b f(x+a)\,dx$
  4. $\int_a^b f(x)\,dx = \int_a^b f(b/x)\,dx$
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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.

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Using the King's property, $\int_0^a f(x)\,dx$ equals:

  1. $\int_0^a f(a-x)\,dx$
  2. $\int_0^a f(a+x)\,dx$
  3. $\int_0^a f(x-a)\,dx$
  4. $-\int_0^a f(x)\,dx$
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Correct answer: A. $\int_0^a f(a-x)\,dx$

Special case of King's property with $b = a$, lower = 0: $\int_0^a f(x)\,dx = \int_0^a f(0+a-x)\,dx = \int_0^a f(a-x)\,dx$.

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Using the King's property, evaluate $\int_0^{\pi/2} \dfrac{dx}{1 + \sqrt[3]{\tan x}}$:

  1. $\pi/4$
  2. $\pi/2$
  3. $\pi$
  4. $\pi/8$
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Correct answer: A. $\pi/4$

Let $I = \int_0^{\pi/2} dx/(1+\sqrt[3]{\tan x}) = \int_0^{\pi/2} \sqrt[3]{\cos x}/(\sqrt[3]{\cos x}+\sqrt[3]{\sin x})\,dx$. **Apply King:** $I = \int_0^{\pi/2} \sqrt[3]{\sin x}/(\sqrt[3]{\sin x}+\sqrt[3]{\cos x})\,dx$. Adding gives $2I = \int_0^{\pi/2} 1\,dx = \pi/2$, so $I = \pi/4$. Beautiful symmetry trick that works for ANY $f$, not just cube root.

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