Which of the following is not correct ?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\int_{0}^{2a} f(x)\,dx=\int_{0}^{a} f(x)\,dx+\int_{0}^{a} f(a-x)\,dx\)
The correct property for the interval [0, 2a] is
\( \displaystyle \int_{0}^{2a} f(x)\,dx = \int_{0}^{a} f(x)\,dx+\int_{0}^{a} f(2a-x)\,dx \)
(split at a and put x = 2a − t in the integral over [a, 2a]). Checking each option against the standard properties:
Option (1) \(\int_{0}^{2a} f(x)\,dx=\int_{0}^{a} f(x)\,dx+\int_{0}^{a} f(a-x)\,dx\) — Incorrect. Since \(\int_{0}^{a} f(a-x)\,dx = \int_{0}^{a} f(x)\,dx\), the right side is \(2\int_{0}^{a} f(x)\,dx\), which fails in general: for f(x) = x, a = 1 the left side is 2 and the right side is 1. So this is the statement that is not correct.
Option (2) \(\int_{a}^{b} f(x)\,dx=\int_{a}^{b} f(a+b-x)\,dx\) — Correct. The substitution x = a + b − t swaps the limits a and b.
Option (3) \(\int_{0}^{2a} f(x)\,dx=2\int_{0}^{a} f(x)\,dx\) if \(f(2a-x)=f(x)\) — Correct. In the property above, the second integral then equals the first.
Option (4) \(\int_{0}^{2a} f(x)\,dx=0\) if \(f(2a-x)=-f(x)\) — Correct. In the property above, the second integral is then the negative of the first.
NCERT (2026-27 edition):
Some Properties of Definite Integrals
