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

**Mattpd** · July 6th 2010, 08:00 PM

$\int (1/1+t^2)i+sec^2(t)j+(te^-^t)k$

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?

what about the third? not sure how to go about it.

**Chris L T521** · July 7th 2010, 02:29 AM

Originally Posted by Mattpd

$\int (1/1+t^2)i+sec^2(t)j+(te^-^t)k$

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?

what about the third? not sure how to go about it.

You integrate the vector valued function component-wise.

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

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

Can you proceed?

**Mattpd** · July 7th 2010, 11:18 AM

I think so.. Is it -te^-t - e^-t

**mr fantastic** · July 7th 2010, 02:32 PM

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).

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