# SOLVED: 2 A function f is called even if f(-x) = f(z) on the domain of f For example, f(x) = x2 and g(z) = cos(x) are even functions A function f is called odd if f(-x) = f(z) o the domain of f For ex

Odd and Even Functions | Integration of odd and even functions
Odd and Even Functions | Integration of odd and even functions

Get 5 free video unlocks on our app with code GOMOBILE

Snapsolve any problem by taking a picture.
Try it in the Numerade app?

Solved step-by-step

2_ A function f is called even if f(-x) = f(z) on the domain of f For example, f(x) = x2 and g(z) = cos(x) are even functions A function f is called odd if f(-x) = ~f(z) o the domain of f _ For example, f(z) = x3 and g(x) = sin() are odd functions. Most functions are: never odd O” even _ a) The integral of an even function over a symmetric interval is twice the integral over half the interval.
f(x) dx = 2
f(x) dx for even f
Prove this formula by using f-R f(x) dx = f’R f(z) dx + Jor f(x) dx and by using integration by substitution on f-R f(x) dx.
b) The integral of an odd function over a symmetric interval is zero
f(x) dx = 0 for odd f ~R R Prove this formula by using ~R f(x) dx = f-R f(r) dx + J” f(r) dx and by using integration by substitution on f’R f(r) dx:
Do not try to prove these formulas by drawing pictures. The problem is to prove these formulas by using integration by substitution:

Solved by verified expert

This problem has been solved!

Try Numerade free for 7 days

03:05

(a) Let $f$ be an odd function; that is, $f(-x)=-f(x) .$ Invent a theorem that makes a statement about the value of an integral of the form $$\int_{-a}^{a} f(x) d x$$(b) Confirm that your theorem works for the integrals $\int_{-1}^{1} x^{3} d x \quad$ and $\quad \int_{-\pi / 2}^{\pi / 2} \sin x d x$(c) Let $f$ be an even function; that is, $f(-x)=f(x) .$ Invent a theorem that makes a statement about the relationship between the integrals$$\int_{-a}^{a} f(x) d x \quad \text { and } \quad \int_{0}^{a} f(x) d x$$(d) Confirm that your theorem works for the integrals$$\int_{-1}^{1} x^{2} d x \quad \text { and } \quad \int_{-\pi / 2}^{\pi / 2} \cos x d x$$

01:23

Let $F$ be any function whose domain contains $-x$ whenever it contains $x$. Prove each of the following.(a) $F(x)-F(-x)$ is an odd function.(b) $F(x)+F(-x)$ is an even function.(c) $F$ can always be expressed as the sum of an odd and an even function.

02:55

A function $f$ is said to be even if $f(-x)=f(x)$ for all $x$ A function $f$ is said to be odd if $f(-x)=-f(x) .$ Suppose that $f$ is continuous for all $x$. Show that if $f$ is even, then $\int_{-a}^{a} f(x) d x=2 \int_{0}^{a} f(x) d x .$ Also, if $f$ is odd, show that $\int_{-a}^{a} f(x) d x=0$

10:16

Recall that a function $f$ is called \textit{even} if $f(-x) = f(x)$ for all $x$ in its domain and \textit{odd} if $f(-x) = -f(x)$ for all such $x$. Prove each of the following.(a) The derivative of an even function is an odd function.(b) The derivative of an odd function is an even function.

09:11

Recall that a function $f$ is called even if $f(-x)=f(x)$ for all $x$ in its domain and odd if $f(-x)=-f(x)$ for all such $x$ . Prove each of the following.(a) The derivative of an even function is an odd function.(b) The derivative of an odd function is an even function.

Transcript

Hello Children given question is we have to throw that integration from integration from minus R two R right ff X multiplied by the X. Right is a cost Oh two times integration from zero to our DX. multiplied by the X. For even if Right. For you win even Yes that means even functioning for effects. And also we worked through that integration from minus our to our cf affects F of Yes multiplied by 80 X is equal to zero for hard value. Well fix where are you function? Yes. Right. By using the formula integration from minus R two R here for affects multiplied by the Xz cost. Oh integration from minus arto zero. Therefore effects multiplied by The ex place integration from 0 to our the F of X. Multiplied by dx by using suitable substitution for minus zero minus R 20 of affects multiplied by the X. Now for this he put the XZ cost. Oh that means the easy Questo minus X Then what you will get dtz questo minus dx at eight then if x is equal to minus R right? If X is ico sto minus are what I will get that in place. The easy cost o r Right? If X is equal to zero The cost zero apply that implies that is equal to zero. Right then integration become minus or two or Yeah, for a fix Multiplied by the XZ coastal integration from our 20 Right R 20 FF right. In the place of out In the place of Yes we can support dynasty rate minus day now. In the place of duty we can support minus ditty Right minus DT place integration from zero to our your father. Yeah,…

Create an account to get free access

or

EMAIL