Odd functions and even functions

Reference. I will refer to the textbook for this class using indicators such as relevant section numbers or page numbers. However, the contents here are NOT directly from the book, so I am not violating any copyright, to my best judgement.

The answer to the last exercise of the previous posting is as follows: $$\int_{-1}^{1} x dx = \int_{-1}^{1}x^{3} dx = \int_{-1}^{1}x^{5} dx = \int_{-1}^{1}x^{7} dx = 0.$$ In general, you may note that $$\int_{-a}^{a}x^{n}dx = 0$$ for any odd positive integer $n$ regardless of what $a$ is. The function $f(x) = x^{n}$ satisfies $f(-x) = -f(x)$ when $n$ is odd. This means that the graph $y = f(x)$ is symmetric with respect to the origin $(0, 0).$ (Can you see why?) Such function (i.e., the one with $f(-x) = -f(x)$) is said to be odd.

Odd continuous function cancellation (p.306). If $f(x)$ is an odd continuous function defined on $[-a, a],$ then $$\int_{-a}^{a}f(x)dx = 0.$$ Why is this useful? Here is an exercise where you can use this:

Exercise. Compute $$\int_{-3000}^{3000}\sin(\sin(x))dx.$$ (Hint: show that $f(x) = \sin(\sin(x))$ is an odd function.)

Q. Can we use this cancellation trick to the integral $$\int_{-1}^{1}x^{2} dx?$$ No! Draw the picture of the graph $y = x^{2}$ and consider the region under the graph from $x = -1$ to $x = 1$. The entire region is above the $x$-axis, so we can see that the answer is positive. However, instead of cancellation, you may notice the "doubling" effect: $$\int_{-1}^{1}x^{2} dx = 2\int_{0}^{1}x^{2} dx.$$ Using 1st FTC, we can quickly compute $\int_{0}^{1}x^{2} dx = 1/3,$ so the answer to the above computation is $2 \cdot (1/3) = 2/3.$

In general, we can see that $$\int_{-a}^{a} x^{n} dx = 2 \int_{0}^{a} x^{n} dx,$$ for any even positive integer $n$ regardless of what $a$ we choose. The function $f(x) = x^{n}$ satisfies $f(-x) = f(x)$ when $n$ is even. This means that the the graph $y = f(x)$ is symmetric with respect to the $y$-axis. (Can you see why?) Such a function (i.e., the one with $f(-x) = f(x)$) is said to be even.

Exercise. Compute $$\int_{-3000}^{3000} e^{|x|} + \sin(2x)e^{\cos(x)} dx.$$ (Hint: first, note that $\sin(2x)e^{\cos(x)}$ is odd. Then note that $e^{|x|}$ is even and also that $|x| = x$ when $x \geq 0.$)