Given f(a-x)=f(x) and f(2 a-x)=-f(x). Also given that...

Given $f(a-x)=f(x)$ and $f(2 a-x)=-f(x)$. Also given that $g(a-x)=-g(x)$ and $g(2 a-x)=g(x)$. Then (a) $\int_{0}^{a} \mathrm{f}(x) \mathrm{g}(\mathrm{x}) \mathrm{d} \mathrm{x}=\int_{0}^{1} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x} \int_{0}^{*} \mathrm{~g}(\mathrm{x}) \mathrm{d} \mathrm{x}$ (b) $\int_{0}^{a} f(x) g(x) d x=0$ (c) $\int_{0}^{2_{2}} f(x) g(x) d x=0$ (d) $\int_{0}^{2 a} f(x) g(x) d x=2 \int_{0}^{2} f(x) g(x) d x$

Given $f(a-x)=f(x)$ and $f(2 a-x)=-f(x)$. Also given that $g(a-x)=-g(x)$ and $g(2 a-x)=g(x)$. Then
(a) $\int_{0}^{a} \mathrm{f}(x) \mathrm{g}(\mathrm{x}) \mathrm{d} \mathrm{x}=\int_{0}^{1} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x} \int_{0}^{*} \mathrm{~g}(\mathrm{x}) \mathrm{d} \mathrm{x}$
(b) $\int_{0}^{a} f(x) g(x) d x=0$
(c) $\int_{0}^{2_{2}} f(x) g(x) d x=0$
(d) $\int_{0}^{2 a} f(x) g(x) d x=2 \int_{0}^{2} f(x) g(x) d x$

Key Concepts

Integration over Symmetric Intervals

This key concept involves evaluating integrals by exploiting the symmetric or antisymmetric nature of functions over intervals that are symmetric with respect to the function's center of symmetry. When a function is odd about a point and is integrated over an interval symmetric about that point, cancellation occurs, often leading to a zero result.

Change of Variables in Integrals

Changing the variable of integration can help align the limits with the symmetry of the functions being integrated. This technique is crucial when dealing with functions that exhibit reflection symmetries, as it often transforms the integral into a form where the odd or even nature of the function becomes evident and simplifies the calculation.

Symmetry of Functions

This concept deals with functions that obey specific reflection properties about a point or line. By having properties such as being invariant or changing sign when evaluated at symmetric points, the analysis of these functions, especially when integrated over symmetric intervals, can be considerably simplified.

Even and Odd Functions

Even and odd functions are special cases of symmetric functions where an even function satisfies f(?x)=f(x) and an odd function satisfies f(?x)=?f(x). When these properties are generalized to other axes by shifting the argument, they continue to provide powerful tools for simplifying integrals, particularly because the integral of an odd function over a symmetric interval is zero.

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