How does knowing a function as even or odd help in integration ??

Solution 1:

Odd and even functions

Even and odd are terms used to describe particularly well-behaved functions.

An even function is symmetric about the $y$-axis. That is, if we reflect the graph of the function in the $y$-axis, then it doesn’t change. Formally, we say that $\,f$ is even if, for all $x$ and $-x$ in the domain of $\,f$, we have $$f(-x)=f(x)$$ Examples of even functions are $\,f(x)=x^2$ and $\,f(x)=\cos x$.

An odd function has rotational symmetry of order two about the origin. That is, if we rotate the graph of the function $180^\circ$ about the origin, then it doesn’t change. Formally, we say that $\,f$ is odd if, for all $x$ and $-x$ in the domain of $\,f$, we have $$f(-x)=-f(x)$$ Examples of odd functions are $\,f(x)=x^3$ and $\,f(x)=\sin x$.

Integration

When calculating Fourier series, you often consider integrals of the form $$I=\int_{-a}^a f(x)\,\mathrm{d}x$$ If $\,f$ is odd or even, then sometimes you can make this simpler. We can rewrite that integral in the following way: \begin{align*} I=\int_{-a}^a f(x)\,\mathrm{d}x &= \int_{-a}^0 f(x)\,\mathrm{d}x + \int_0^a f(x)\,\mathrm{d}x \\ &= \int_0^a f(-x)\,\mathrm{d}x + \int_0^a f(x)\,\mathrm{d}x \end{align*} For an even function, we have $f(-x)=f(x)$, whence $$I = 2\int_0^a f(x)\,\mathrm{d}x$$ For an odd function, we have $f(-x)=-f(x)$, whence $$I = \int_0^a (-f(x)+f(x))\,\mathrm{d}x = 0$$ That’s what it means to simplify the integration: the integral of an odd or even function over the interval $[-a,a]$ can be put into a nicer form (and sometimes we can see that it vanishes without ever computing an integral).

Solution 2:

An even function is the one which satisfies the folloving conditions: 1) for any $x$ from the domain of $f$ the point $-x$ is also in the domain of $f$ 2) $f(-x) = f(x)$ for any $x$ from the domain of $f$.

An odd function is the one that satisfies the conditions 1) for any $x$ from the domain of $f$ the point $-x$ is also in the domain of $f$ 2) $f(-x) = -f(x)$ for any $x$ from the domain of $f$.

The graph of an even function is symmetric with respect to the Y-axis, and the graph of an odd function is symmetric with respect to the origin (0,0). Hope that'll be helpful...