Question

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?

Answer 1

the fact that the derivative of x^5 is present

Answer 2

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

Answer 3

5x^4

Answer 4

not quite. But that does stand out.

Answer 5

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

Answer 6

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.

Answer 7

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

Answer 8

which isn't as nice.

Answer 9

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

Answer 10

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

Answer 11

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

Answer 12

du = 5x^4 ??

Answer 13

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

Answer 14

u = x^5

Answer 15

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

Answer 16

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

Answer 17

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

Answer 18

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

Answer 19

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

Answer 20

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

Answer 21

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

Answer 22

or reverse it 1/5x^4 first

Answer 23

Neither..

Let \(u = x^5 \implies du = 5x^4 dx \implies dx = \frac{1}{5x^4}du\)
\[\int 5x^4e^{x^5}dx = \int(5x^4e^u )\frac{1}{5x^4}du\]

Answer 24

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

Answer 25

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

Answer 26

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

Answer 27

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

Answer 28

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

Answer 29

Does that make sense?

Answer 30

yes it does

Answer 31

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

Answer 32

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

Answer 33

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

Answer 34

Umm.. close

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

Answer 35

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

Answer 36

Right.

Answer 37

so now will I replace u with x^5

Answer 38

After you integrate then you replace it back.

Answer 39

err take the anti-derivative.

Answer 40

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

Answer 41

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

Answer 42

Yep.

Answer 43

good!!!!!
Thanks gotta go now

Answer 44

Though you should slap some parens in there for readability.

Answer 45

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

Answer 46

thanks