Evaluate
aniti-derivitive
5x^4 e^x^5 dx
I know I need to get the u and du, but what about this problem tells me to do that?

- anonymous

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- anonymous

the fact that the derivative of x^5 is present

- anonymous

This one is a bit less obvious, but what part of this equation seems like a good candidate for a u sub?

- anonymous

5x^4

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## More answers

- anonymous

not quite. But that does stand out.

- anonymous

I'm looking at the \(e^{x^5}\) as being particularly ugly, and I'd like to have it be something nice like \(e^u\)

- anonymous

You'll find that in the beginning, you are mostly just guessing at things to pick for u and seeing if they work out nicely. As you get more practice you'll be able to spot things better.

- anonymous

So you can certainly try working with the x^4 as u, but then you'll have \(e^{ux}\)

- anonymous

which isn't as nice.

- anonymous

And you'll have \(5u/x^3\) in front because your du will be \(4x^3 dx\)

- anonymous

So that's a lot of mixed x's and u's.

- anonymous

Which often (especially in the beginning) means you're on the wrong track.

- anonymous

du = 5x^4 ??

- anonymous

When you are first learning u substitution picking a good u, should simplify the problem a lot. What did you pick for u?

- anonymous

u = x^5

- anonymous

ok good. Yes. So what is dx in terms of du ?

- anonymous

du = \(5x^4 dx \implies dx = ?\)

- anonymous

not sure I want to think I take derivative of 5x^4 ??? not sure

- anonymous

No, just divide.
\[du = 5x^4 dx \implies dx = \frac{1}{5x^4} du\]

- anonymous

So now we have something we can plug in for dx and it'll cancel nicely with the product of 5x^4 out front.

- anonymous

so will I put e^u * 1/5x^4

- anonymous

Don't forget the 5x^4 you have in front of the e^u from the initial equation.

- anonymous

or reverse it 1/5x^4 first

- anonymous

Neither..
Let \(u = x^5 \implies du = 5x^4 dx \implies dx = \frac{1}{5x^4}du\)
\[\int 5x^4e^{x^5}dx = \int(5x^4e^u )\frac{1}{5x^4}du\]

- anonymous

Do you follow that and understand where each piece came from?

- anonymous

I don't really understand why 5x^4 stayed in front

- anonymous

Where should it have gone? It's part of the equation, I can't make it evaporate ;)

- anonymous

All I did was substitute u for x^5 and replaced dx with my expression with du.

- anonymous

But I can't do anything to the 5x^4 yet, because that's not x^5 = u.

- anonymous

Does that make sense?

- anonymous

yes it does

- anonymous

But when we do that, we get something nice for our new version that should be easier to take the anti-derivative of.

- anonymous

?? (5x^4 * 1/u *e^u) * 1/5x^4 du

- anonymous

never mind th 1/u should be just 1/1 shouldn't it

- anonymous

Umm.. close
\[\int (5x^4e^u)\frac{1}{5x^4}du = \int e^u\frac{5x^4}{5x^4}du = \int e^u du\]

- anonymous

I can see that because it is all multipilcation no + or -

- anonymous

Right.

- anonymous

so now will I replace u with x^5

- anonymous

After you integrate then you replace it back.

- anonymous

err take the anti-derivative.

- anonymous

Sorry, later on they're going to tell you that anti-derivatives are called integrals. ;p

- anonymous

so will the answer be e^x^5 + c

- anonymous

Yep.

- anonymous

good!!!!!
Thanks gotta go now

- anonymous

Though you should slap some parens in there for readability.

- anonymous

when you type it that is. I'm sure it's written right on your paper.

- anonymous

thanks

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