## anonymous 5 years ago 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?

1. anonymous

the fact that the derivative of x^5 is present

2. anonymous

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

3. anonymous

5x^4

4. anonymous

not quite. But that does stand out.

5. anonymous

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

6. 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.

7. anonymous

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

8. anonymous

which isn't as nice.

9. anonymous

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

10. anonymous

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

11. anonymous

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

12. anonymous

du = 5x^4 ??

13. anonymous

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

14. anonymous

u = x^5

15. anonymous

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

16. anonymous

du = $$5x^4 dx \implies dx = ?$$

17. anonymous

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

18. anonymous

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

19. 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.

20. anonymous

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

21. anonymous

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

22. anonymous

or reverse it 1/5x^4 first

23. anonymous

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

24. anonymous

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

25. anonymous

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

26. anonymous

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

27. anonymous

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

28. anonymous

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

29. anonymous

Does that make sense?

30. anonymous

yes it does

31. anonymous

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

32. anonymous

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

33. anonymous

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

34. anonymous

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

35. anonymous

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

36. anonymous

Right.

37. anonymous

so now will I replace u with x^5

38. anonymous

After you integrate then you replace it back.

39. anonymous

err take the anti-derivative.

40. anonymous

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

41. anonymous

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

42. anonymous

Yep.

43. anonymous

good!!!!! Thanks gotta go now

44. anonymous

Though you should slap some parens in there for readability.

45. anonymous

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

46. anonymous

thanks