Help with Calculus problem?

Hey, an odd function is one that is symmetric with respect to the origin, i.e. f(−x) = −f(x) for all x in the domain. Some examples are f(x) = x³ and f(x) = sin(x). Make sure the definition makes sense by plugging in some numbers for yourself, and looking at their graphs! (Alternatively, an even function is one symmetric with respect to the y-axis, i.e. f(−x) = f(x) for all x in the domain. Some examples are f(x) = x² and f(x) = cos(x). Additionally, some functions are neither even nor odd.)

To do this proof, we first split up the integral into 2 parts:

∫ f(x)dx [from −a to a] = ∫ f(x)dx [from −a to 0] + ∫ f(x)dx [from 0 to a]

Now we look at this first half (the integral from −a to 0), and do some substitution using the substitution

u = −x

Thus, du = −dx

∫ f(x)dx [from −a to 0]

Doing our substitution, we replace x with −u and dx with −du. Remember also to substitute the ends of the integral (u = −x, so −a goes to a and 0 goes to −0 = 0).

= ∫ f(−u)(−du) [from a to 0]

Switching the ends of the integral gets rid of the minus sign.

= ∫ f(−u)du [from 0 to a]

Rename the "dummy" integration variable from u to x.

= ∫ f(−x)dx [from 0 to a]

Since f is an odd function, remember f(−x) = −f(x).

= ∫ −f(x)dx [from 0 to a]

Take the minus sign outside the integral.

= − ∫ f(x)dx [from 0 to a]

Now we can go back to our original statement,

∫ f(x)dx [from −a to a] = ∫ f(x)dx [from −a to 0] + ∫ f(x)dx [from 0 to a]

Replace the first half with what we just calculated and simplify.

∫ f(x)dx [from −a to a]

= − ∫ f(x)dx [from 0 to a] + ∫ f(x)dx [from 0 to a]

= 0

I hope that helped, and also thanks for your message! It means a lot that you remembered me :)

An odd function is one which displays rotational symmetry around the origin, for example, any y=x^n where n is odd, or any number of x terms which all have odd powers, hence including the sine function, the tan function and many more. This means f(a) = -f(-a). For an even function, f(a) = f(-a). An even function is y=x^n where n is even, cosine, and has a line of symmetry in the y axis. So, when considering the integration, since we know the power rule of integration:

Integrating x^n + x^m+ … dx where n is odd will yield 1/n+1 x^n+1 + 1/m+1 x^m+1 +C. These terms will all have an even exponent, and so the integral of f(x) is even. Call the integral of f(x) "g(x)". Then, between the limits of a and -a, we will have: g(a) - g(-a). Since we know g(a) = g(-a) from our definition of even functions, g(a) -g(-a) = 0. Hope this helps!