Question

Evaluate the integral using the properties of even and odd functions as an aid.

Answer 1

Well, do it. Is it even or odd? Is the range of integration symmetrical about the origin?

Answer 2

I'm pretty sure it's odd.

Answer 3

Please write the definition of an Odd Function. Pretty sure isn't good enough,

Answer 4

Even: f(x) = f(-x) Is that true for this function?
Odd: f(x) = -f(-x) Is that true for this function?

Answer 5

\[\int\limits_{-2}^{2} x ^{2}(x ^{2}+1) dx\]

Answer 6

Help please

Answer 7

You haven't answer the question. Is it Odd or Even? Check out those definitions.

Answer 8

Oh its even

Answer 9

How do you know?

Answer 10

An even function is a function such that f(-x)=f(x). Graphically an even function is symmetrical about the y-axis while an odd function is symmetrical when rotated about the origin 180 degrees.

Answer 11

Am I right?

Answer 12

Let's see: \(f(-x) = (-x)^{2}((-x)^{2}+ 1) = x^{2}(x^{2}+1)= f(x)\) -- Perfect.

Okay, how do we utilize this to simplify the integral?

Answer 13

hold on

Answer 14

do you use\[u = x ^{2}+1 \]
which is du is 2x

Answer 15

while dx is du/2x

Answer 16

How does that utilize the fact that it is an Even Function?

First, there is no need for a substitution. This is a simple integral.

The lesson, here, is about symmetry.

Answer 17

Im confused?

Answer 18

The point you are to see is that we have this situation:

1) Even Function
2) Symmetric About the y-Axis
3) Half the integral is redundant.

\(\int\limits_{-2}^{2}x^{2}(x^{2}+ 1)\;dx = 2\cdot\int\limits_{0}^{2}x^{2}(x^{2}+ 1)\;dx\)

Answer 19

oh yeah.. Now I got it

Answer 20

after this do you add 1

Answer 21

Sorry, mom called me

Answer 22

Add 1 for what? Just evaluate the integral.

Answer 23

okay, I'll let me work on it as soon as possible. I'm eating dinner

Answer 24

I had Frog Legs and Pork Chops. It was pretty good.

Answer 25

lol

Answer 26

That's cool

Answer 27

I don't eat meat

Answer 28

Also cool.

Answer 29

I heard people eat all sorts of thing. None of them make me scared

Answer 30

Well, let's eat some calculus, shall we?

Answer 31

lol Im almost done ... let me drink water

Answer 32

My frogs would like that.

Answer 33

I find it little hard writing the equation down

Answer 34

\[2 \int\limits_{0}^{2} x^{2}(x ^{2}+1)dx \]

Answer 35

\[x ^{2} (x ^{2}+1) \int\limits_{0}^{5}\]

Answer 36

Is this something like this

Answer 37

Remember, this is a trivial integral. If you are thinking of something fancy, you don't see it.

Answer 38

wouldn't you subtitute (2^2 +1) = 5 and 0^2+ 1 which the bottom is 1

Answer 39

instead of 0. If I'm wrong help?

Answer 40

Am I right?

Answer 41

Umm... You should first find the anti-derivative.

\(\int x^{2}(x^{2} + 1)\;dx = \int x^{4} + x^{2}\;dx = ??\)

Answer 42

\[\frac{ x }{ 5}^{5}+ x ^{2}\]

Answer 43

right?

Answer 44

I mean x^ 3

Answer 45

\[\frac{ 1 }{ 5 }x ^{5}+ x ^{3}\]

Answer 46

Why didn't we get \(\dfrac{1}{5}x^{5} + \dfrac{1}{3}x^{3} + C\)

Answer 47

oh yeah plus C

Answer 48

I think you made a mistake its x cube

Answer 49

never mind sorry

Answer 50

What do you do after this? do you subtitute the 2 and 0 to get the answer for this question

Answer 51

meanwhile you subtract both of them right

Answer 52

First, you get the integral right. If you think there is no 1/3 in there, you should review your notes. Why did you get a 1/5 on the first piece, but nothing on the second piece? That makes no sense.

Answer 53

Well sorry.. wasn't paying attention.

Answer 54

I don't know what is wrong with me

Answer 55

I hope you understand me.. I make lot of mistake..

Answer 56

I need to get this done because I have headache.. hope you understand right

Answer 57

I just had seizure recently..

Answer 58

It is okay to make mistakes so long as you learn from them.

\(2\cdot\int\limits_{0}^{2} x^{4} + x^{2}\;dx = \dfrac{1}{5}x^{5} + \dfrac{1}{3}x^{3}|_{0}^{2} = 2*(\left((1/5)32+(1/3)8\right)-\left(0\right)) = 2(35/5 + 8/3)\)

Answer 59

did you got 32 by 3^5

Answer 60

sorry 2^5

Answer 61

I caught up

Answer 62

is it 9\[9 \frac{ 2 }{ 3 }\]

Answer 63

?? Sorry, typo up there. 32, not 35

2(32/5 + 8/3) = 16(4/5 + 1/3) = 16(12/15 + 5/15) = 16 (17/15)

It's just arithmetic.

Answer 64

Am I right to the final answer?

Answer 65

You should be able to compare my correct response with yours. You tell me if yours is correct.

Answer 66

it's wrong... but I should be multiplying by 2

Answer 67

in the end

Answer 68

Thanks for help!

Answer 69

Did you follow mine at all? The only error was that goofey 35 and the cut off of the final /3.

Down below, all the arithmetic is shown. Keep trying until you get 16 (17/15) = 272/15

Answer 70

alright I got it
I did it.

Answer 71

Thanks for help!