# (Vector calc) how do you integrate this vector function?

• Jul 6th 2010, 08:00 PM
Mattpd
(Vector calc) how do you integrate this vector function?
$\displaystyle \int (1/1+t^2)i+sec^2(t)j+(te^-^t)k$

How do you treat i, j, and k? what about t? I assume you break this into three integrals, and I think you pull out the i, j, and k, but what from there?

Is the first integral arctan(t)i
and the second tan(t)j?

• Jul 7th 2010, 02:29 AM
Chris L T521
Quote:

Originally Posted by Mattpd
$\displaystyle \int (1/1+t^2)i+sec^2(t)j+(te^-^t)k$

How do you treat i, j, and k? what about t? I assume you break this into three integrals, and I think you pull out the i, j, and k, but what from there?

Is the first integral arctan(t)i
and the second tan(t)j?

You integrate the vector valued function component-wise.

So yes, you will have $\displaystyle \displaystyle (\arctan t) \mathbf{i}+ (\tan t ) \mathbf{j}+ \left( \int te^{-t} \, dt \right) \mathbf{j} + \mathbf{C}$, where $\displaystyle \mathbf{C}$ is a constant vector.

To integrate $\displaystyle \displaystyle\int te^{-t}dt$, use integration by parts.

Can you proceed?
• Jul 7th 2010, 11:18 AM
Mattpd
I think so.. Is it -te^-t - e^-t
• Jul 7th 2010, 02:32 PM
mr fantastic
Quote:

Originally Posted by Mattpd
I think so.. Is it -te^-t - e^-t

You can always check the answer to an integration by differentiating it. Do you get back te^(-t) when you do this? (And normally I'd say there's a "+ C" but that has already been taken into account in Chris's answer).