Find the indefinite integral (xe^x)dx

Question

zehanz · Answer

Do you know Integration by Parts?

anonymous · Answer

Not really. We just went over it, but I didn't understand.

zehanz · Answer

I mean this rule: $\int fG=FG-\int Fg$, whre F'=f and G'=g

anonymous · Answer

No

zehanz · Answer

Without it, you cannot find this one...are you sure? It is impossible otherwise.

anonymous · Answer

Well my teacher went over that rule, I just don't understand how to use it.

zehanz · Answer

OK, let my give it a try!

If you look at the rule, it tells you what to do if you have an integral of a function that is a product of two other functions.
You do have this situation here, because your function is $x\cdot e^{x}$. Right?

zehanz · Answer

At first, it doesn't seem to help much, because the rule tells you to replace one integral with FG - another integral.

anonymous · Answer

yes, but I was looking at my notes and she never taught us the rule you had above. She taught us ∫udv = uv-∫vdu. Is that the same thing?

zehanz · Answer

The trick is, you may choose f and G yourself, and then in such a way that the second integral CAN easily be done.

I know this sounds complicated, but it really isn't.

Let me choose the functions for you:

f(x)= e^x, then G(x)=x.

Can you tell me what g(x) is? (remember, g=G')

zehanz · Answer

BTW, your rule is the same as mine, only in a slightly different notation.

anonymous · Answer

isn't it just 1dx?

zehanz · Answer

It is! And it is simple!

Can you give me F, if f(x)=e^x?

anonymous · Answer

e^x?

zehanz · Answer

Right again! We're almost done.

We only have to plug in these thing in the rule:

$\int xe^{x}dx=xe^x-\int1\cdot e^xdx=$, so

$\int xe^{x}dx=xe^x-\int e^xdx$.

Can you do the last step?

anonymous · Answer

xe^x-e^x and then I add +c to the end for these equations right?

zehanz · Answer

Yes! You are a winner!

anonymous · Answer

I guess that wasn't as hard as I thought it was haha. Thank you so much!

zehanz · Answer

YW! Please remember the rule of integration by parts :D

zehanz · Answer

You might be needing it again some time....

anonymous · Answer

I'll try haha. I actually have two more questions. Can you help me with them?

zehanz · Answer

Just ask them as new questions, so anyone can have a go at them.