anonymous
  • anonymous
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?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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anonymous
  • anonymous
the fact that the derivative of x^5 is present
anonymous
  • anonymous
This one is a bit less obvious, but what part of this equation seems like a good candidate for a u sub?
anonymous
  • anonymous
5x^4

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anonymous
  • anonymous
not quite. But that does stand out.
anonymous
  • 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
  • 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
  • anonymous
So you can certainly try working with the x^4 as u, but then you'll have \(e^{ux}\)
anonymous
  • anonymous
which isn't as nice.
anonymous
  • anonymous
And you'll have \(5u/x^3\) in front because your du will be \(4x^3 dx\)
anonymous
  • anonymous
So that's a lot of mixed x's and u's.
anonymous
  • anonymous
Which often (especially in the beginning) means you're on the wrong track.
anonymous
  • anonymous
du = 5x^4 ??
anonymous
  • 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
  • anonymous
u = x^5
anonymous
  • anonymous
ok good. Yes. So what is dx in terms of du ?
anonymous
  • anonymous
du = \(5x^4 dx \implies dx = ?\)
anonymous
  • anonymous
not sure I want to think I take derivative of 5x^4 ??? not sure
anonymous
  • anonymous
No, just divide. \[du = 5x^4 dx \implies dx = \frac{1}{5x^4} du\]
anonymous
  • 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
  • anonymous
so will I put e^u * 1/5x^4
anonymous
  • anonymous
Don't forget the 5x^4 you have in front of the e^u from the initial equation.
anonymous
  • anonymous
or reverse it 1/5x^4 first
anonymous
  • 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
  • anonymous
Do you follow that and understand where each piece came from?
anonymous
  • anonymous
I don't really understand why 5x^4 stayed in front
anonymous
  • anonymous
Where should it have gone? It's part of the equation, I can't make it evaporate ;)
anonymous
  • anonymous
All I did was substitute u for x^5 and replaced dx with my expression with du.
anonymous
  • anonymous
But I can't do anything to the 5x^4 yet, because that's not x^5 = u.
anonymous
  • anonymous
Does that make sense?
anonymous
  • anonymous
yes it does
anonymous
  • 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
  • anonymous
?? (5x^4 * 1/u *e^u) * 1/5x^4 du
anonymous
  • anonymous
never mind th 1/u should be just 1/1 shouldn't it
anonymous
  • 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
  • anonymous
I can see that because it is all multipilcation no + or -
anonymous
  • anonymous
Right.
anonymous
  • anonymous
so now will I replace u with x^5
anonymous
  • anonymous
After you integrate then you replace it back.
anonymous
  • anonymous
err take the anti-derivative.
anonymous
  • anonymous
Sorry, later on they're going to tell you that anti-derivatives are called integrals. ;p
anonymous
  • anonymous
so will the answer be e^x^5 + c
anonymous
  • anonymous
Yep.
anonymous
  • anonymous
good!!!!! Thanks gotta go now
anonymous
  • anonymous
Though you should slap some parens in there for readability.
anonymous
  • anonymous
when you type it that is. I'm sure it's written right on your paper.
anonymous
  • anonymous
thanks

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