Question

Find the indefinite integral of [e^(-t)sin(t)i + e^(-t)cos(t)j] dt. side note: I'm not entirely sure of what to do with e^(-t)....

Answer 1

please use equation editor so that I can get a better look of the integration problem...thanks

Answer 2

\[\int\limits_{}^{} [e ^{-t}\sin(t)i + e ^{-t}\cos(t)j] dt\]

Answer 3

sorry, didn't know that existed haha.

Answer 4

Well this is an integration by parts problem or you can use that the imaginary part of e^(i t) is sin(t) and the real part of e^(it) is cos(t).

Answer 5

And I assume those are i and j unit vectors?

Answer 6

yes they are

Answer 7

So how would you integrate:
\[\int\limits e^{-t} \cos(t)dt\]?

Answer 8

\[\frac{ 1 }{ 2 }e ^{-t}(\sin(t)-\cos(t))+C \] ? i think...

Answer 9

Differentiate it and see if it works:
\[\frac{1}{2}\frac{d}{dt} \left( e^{-t}\sin(t) -e^{-t}\cos(t) \right) = \frac{1}{2} \left( -e^{-t}\left(\sin(t) - \cos(t) \right) + e^{-t}(\cos(t) + \sin(t)\right)\]
\[=\frac{1}{2}(2 e^{-t} \cos(t)) = e^{-t}\cos(t)\]
This confirms your integral is correct. So then do the other one involving sine. From there you know that:
\[\int\limits \left( f(x) \hat{i} + g(x) \hat{j}\right)dx = \left( \int\limits f(x) dx \right) \hat{i} + \left(\int\limits g(x) dx \right) \hat{j}\]

Answer 10

So simply integrate and then add on a constant vector C which encompasses the constant of integration you get from integrating the x component (the one times i hat) and the one you get from integrating the y component (the one times j hat).

Answer 11

\[\frac{ 1 }{ 2 }e ^{-t}(\sin(t)-\cos(t))i + ((\frac{ -1 }{ 2 }e ^{-t}(\sin(t)+\cos(t))j) + C\]
?

Answer 12

Yes but remember that C should be:
\[\vec{C} = c_1 \hat i + c_2 \hat j\]
since both the x and y components have a constant. (If you want to know how to do this in latex simply type \vec{whatever}, this will give:
\[\vec{whatever}\]

Answer 13

I put that in, but I still got it wrong D:

Answer 14

on the site that I turn it in, the C is already included so I only have to type in the previous part of the answer...is there something I'm missing still?

Answer 15

http://www.wolframalpha.com/input/?i=integral+of+e%5E%28-t%29sin%28t%29
The x component is right

http://www.wolframalpha.com/input/?i=integral+of+e%5E%28-t%29cos%28t%29
Your y component is wrong. You have a negative where it shouldn't be.

Answer 16

wait the -1/2 ?

Answer 17

nevermind, I got it. Thank you so so much!!