integrate |cos(e^t)|

Question

anonymous · Answer

$$\int\limits_{0}^{2.5} |\cos(e^t)|$$

anonymous · Answer

Normally I could do one of these with u-substitution easy but this one is really hard.

australopithecus · Answer

yes it is piece wise integration you have two integrals

australopithecus · Answer

$$\int\limits_{0}^{2.5}-\cos(e^{t})$$
and
$$\int\limits_{0}^{2.5}\cos(e^{t}) $$

figure out where cos(e^(t)) = 0 to define which x values belong to which one of the two integrals

australopithecus · Answer

remember an absolute value can only output positive numbers

anonymous · Answer

It equals zero in lots of places: http://www.wolframalpha.com/input/?i=cos%28e%5Et%29+%3D+0

australopithecus · Answer

You can just take the integral of 

$$\int\limits_{0}^{2.5} \cos(e^{t})dt$$

as 0 to 2.5 is a positive out put of cos(e^(t))

australopithecus · Answer

thus the answer is

-sin(e^(2.5)) - sin(e^(0))

australopithecus · Answer

Although I'm probably wrong

anonymous · Answer

Hmm..that's not turning out right for my question options.

amistre64 · Answer

you have no e^t on the outside of it to integrate by parts

amistre64 · Answer

try a sin power series

australopithecus · Answer

oh yeah I integrated it wrong

australopithecus · Answer

cos(e^(t))
u = e^(t)
du = e^(t)dt
du/e^(t) = dt

amistre64 · Answer

err, cos power series, lost track of the post lol

amistre64 · Answer

http://www.wolframalpha.com/input/?i=integrate+%7Ccos%28e%5Et%29%7C eww

australopithecus · Answer

this integral is really ugly :)

amistre64 · Answer

u = e^t

du = e^t dt

du/u = dt

australopithecus · Answer

oh :)

amistre64 · Answer

but yeah, i think a numerical approach might be simplest

anonymous · Answer

I can't think of anything that multiplied by e^t will give you cos(e^t)

amistre64 · Answer

yout not going to find any elementary functions for this

amistre64 · Answer

is your post correct?  check for typos

amistre64 · Answer

http://www.wolframalpha.com/input/?i=y+%3De%5E%282t%29%2F2%21+-e%5E%284t%29%2F4%21+%2Be%5E%286t%29%2F6%21+-e%5E%288t%29%2F8%21+%2Be%5E%2810t%29%2F10%21+-e%5E%2812t%29%2F12%21+and+y%3Dcos%28e%5Et%29 that was pretty cool, i made a flip of it

anonymous · Answer

The post I have of the screen shot should be correct.

amistre64 · Answer

i see it, yeah, numerical analysis would prolly be best

amistre64 · Answer

$$\lim_{n\to\ inf}\ \sum_{i=1}^{n}\frac{2.5}{n}cos(e^{\frac{2.5}{n}i})$$

amistre64 · Answer

http://www.wolframalpha.com/input/?i=sum+2.5%2F1000+cos%28e%5E%282.5n%2F1000%29%29+from+1+to+1000

amistre64 · Answer

ugh, i forgot to determine the |  | parts lol

amistre64 · Answer

1.3937 ... http://www.wolframalpha.com/input/?i=sum+%7C2.5%2F100000+cos%28e%5E%282.5n%2F100000%29%29%7C+from+1+to+100000