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NDA 2023 Maths · Mathematics · Definite Integrals · medium

Consider the following two claims about definite integrals over [0,1]: 1. ∫0^1x dx=∫0^1x^2 dx. 2. ∫0^1x^2 dx=∫0^1(1-x)^2 dx.

Single correct option

Consider the following two claims about definite integrals over [0,1][0,1]:
1. ∫01x dx=∫01x2 dx\displaystyle\int_0^1x\,dx=\int_0^1x^2\,dx.
2. ∫01x2 dx=∫01(1−x)2 dx\displaystyle\int_0^1x^2\,dx=\int_0^1(1-x)^2\,dx.
Which of the statements given above is/are correct?
  1. A. 1 only
  2. B. 2 only
  3. C. Both 1 and 2
  4. D. Neither 1 nor 2
Show answer and solution

Answer: (B) 2 only

The first integrals are 12\dfrac12 and 13\dfrac13, so Statement 1 is false. By the property ∫0af(x)dx=∫0af(a−x)dx\int_0^af(x)dx=\int_0^af(a-x)dx with f(x)=x2f(x)=x^2 and a=1a=1, the second pair are equal (both 13\dfrac13), so Statement 2 is true.
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Original question written in the NDA 2023 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake