Question

Is the following true:

*I will post the mathematical expression as the first comment.

Any and all help is greatly appreciated!

Answer 1

My question:
\[\int\limits_{-\pi/4}^{\pi/4}(x^3+x^4\tan(x))dx = \int\limits_{-\pi/4}^{\pi/4}(u*\tan(u^{1/4})*du/4) \]
if u = x^4. ???

Answer 2

HI!!

Answer 3

Hey misty:)

Answer 4

i wouldn't think so
how did the sum become a product?

Answer 5

Damn.. I thought that might be it. So, would I distribute the dx, and then solve for dx in the du=4x^3*dx part?

Answer 6

btw the answer is pretty clear without finding any anti derivative, which is almost impossible

Answer 7

I thought about factoring out an x^3... is that it? :/

Answer 8

i think the point of this question is something else entirely

Answer 9

look carefully
notice anything about the limits of integration?

Answer 10

let me put in another way
you are not going to find a closed form for the anti derivative, ie. you are not going to find some function whose derivative is the integrand
it is testing something else entirely

Answer 11

it's a half-pi interval, centered on 0? :/ I'll admit, I'm lost right now...

Answer 12

want a hint?

Answer 13

:( Please!

Answer 14

the "centered at zero " part is good

Answer 15

what kind of function is this?
hint, compute \[f(\frac{\pi}{4})\] and \[f(-\frac{\pi}{4})\]

Answer 16

ok i will stop torturing you
the function is odd

Answer 17

OMG!! My professor mentioned that about 2 weeks ago.. vaguely. I see where you're going now!! What's the definition of an odd function?

Answer 18

\[f(-x)=-f(x)\]

Answer 19

Ah.... so it's just sort of an "understand how these work, and look for the hints when they show up" situation?

Answer 20

which is clear because \(x^3\)is odd and \(tan(x)\) is odd and \(x^4\) is even, so \[(-x)^3+(-x)^4\tan(-x)=-x^2-x^4\tan(x)=-(x^3+x^4\tan(x))=-f(x)\]

Answer 21

you integrate an odd function \[\int_{-a}^af(x)dx\]you get \(0\) because of the symmetry

Answer 22

Goddamn you're blowing my mind, and I love it. Gotta... Like, it's reflected around the x-axis because it's Tangent, and since it's from |x| on either side of the y-axis, the two parts are equal to each other, just one's negative? :D

Answer 23

So the answer is 0? :D

Answer 24

yes, definitely zero
think about what it means for a function to be odd

Answer 25

Thank you so much for your help:)

Answer 26

It's possible to get that algebraically though, with u-sub right?

Answer 27

area above equals area below
actually easiest to see with simplest odd function \(f(x)=x\)

Answer 28

no

Answer 29

forget it

Answer 30

... what do I put as "work", just a little not that it's an odd function?

Answer 31

look, just because you can write down a function doesn't mean you can find its anti derivative
it is almost never the case that you can

Answer 32

all that calc 2 stuff you are doing is just showing off
mostly the anti derivative does not have a closed form,
the "work" is to say "function is odd, interval is from \(-a\) to \(a\), answer is 0"

Answer 33

Bueno, sounds good to me:)

Answer 34

that was the entire point of this question, to see if you would realize that

Answer 35

\[\color\magenta\heartsuit\]

Answer 36

♡ :)