Definite Integration
Integral Calculus-2
star_batch_jee_advanced_2025
Grade None
Question:
If $f(x)$ is an even function, then:
$\phi(x)$ is also an even function
$\phi(x)$ is an odd function
If $f(a-x) = -f(x)$, then $\phi(x)$ is an even function
If $f(a-x) = -f(x)$, then $\phi(x)$ is an odd function
Step-by-Step Solution
Key Concept: $\phi(x)$ is the composition of an even function with integration properties that force it to be odd.
For an even function $f(x)$, we have $\phi(-x) = -\int_{-a}^{-x} f(t)dt - \int_a^x f(t)dt = -2\int_0^a f(t)dt - \int_a^x f(t)dt$. Using the substitution property and the fact that $\int_0^a f(a-x)dx = \int_0^a f(x)dx$, we can show that $\int_0^a f(t)dt = 0$, which implies $\phi(-x) = -\int_a^x f(t)dt = -\phi(x)$. Therefore, $\phi(x)$ is an odd function.
Correct Answer: 4