Question

For a function $f(x)$, which of the following holds true?

• A. $\int_a^b f(x) dx = \int_a^b f(a + b - x) dx$
• B. $\int_a^b f(x) dx = 0$, if $f$ is an even function
• C. $\int_a^b f(x) dx = 2 \int_0^a f(x) dx$, if $f$ is an odd function
• D. $\int_0^a f(x) dx = \int_0^a f(2a + x) dx$

Hint:

Always remember the symmetry property of definite integrals: $\int_a^b f(x) dx = \int_a^b f(a + b - x) dx$.

Answer 1

The Correct Option is A

The correct formula is: \[ \int_a^b f(x) dx = \int_a^b f(a + b - x) dx \] This property is known as the symmetry property of definite integrals. It works because when you perform a substitution, $x' = a + b - x$, the limits of integration remain unchanged, and the value of the integral remains the same.