Let f(x) be continuous in R Then, ∫0^21 f(x) d x=2 ∫0^1 f(x) d x is true if (a) f(x) is odd (b)

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Let f(x) be continuous in R Then, ∫0^21 f(x) d x=2 ∫0^1 f(x)...

Let $\mathrm{f}(x)$ be continuous in $\mathrm{R}$ Then, $\int_{0}^{21} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}=2 \int_{0}^{1} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}$ is true if (a) $\mathrm{f}(\mathrm{x})$ is odd (b) $\mathrm{f}(\mathrm{x})$ is symmetric about $\mathrm{x}=\mathrm{a}$ (c) $\mathrm{f}(\mathrm{x})$ is periodic with period a (d) $f(2 a-x)=f(x)$

Let $\mathrm{f}(x)$ be continuous in $\mathrm{R}$ Then, $\int_{0}^{21} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}=2 \int_{0}^{1} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}$ is true if
(a) $\mathrm{f}(\mathrm{x})$ is odd
(b) $\mathrm{f}(\mathrm{x})$ is symmetric about $\mathrm{x}=\mathrm{a}$
(c) $\mathrm{f}(\mathrm{x})$ is periodic with period a
(d) $f(2 a-x)=f(x)$

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Key Concepts

Periodic Functions

A function is periodic with period a if it satisfies f(x + a) = f(x) for all x. This periodicity property implies that the behavior of the function repeats over intervals of length a. In the context of definite integrals, if the interval of integration spans an integer multiple of the period, the integral over the entire interval can often be expressed as a multiple of the integral over one period, leading to potential simplifications.

Odd Functions

An odd function is defined by the property that f(?x) = ?f(x) for all x in its domain. When integrating an odd function over an interval symmetric about the origin—such as [?L, L]—the contributions from symmetric points cancel out, resulting in an integral value of zero. This characteristic is important when analyzing integral properties where symmetry plays a role.

Reflection Symmetry

This concept refers to a function being invariant under a reflection about a specific vertical line. In mathematical terms, a function f is said to be symmetric about the line x = a if it satisfies the condition f(2a - x) = f(x) for all x. Such symmetry often allows one to relate integrals over intervals that are mirror images of each other with respect to the line of symmetry, thereby enabling simplifications of definite integrals.

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