## anonymous 5 years ago Differentiate: 1/(2x-1)^(1/2) dx from x=5 to x=13

1. anonymous

Use a u sub, and then F(13)-F(5)

2. Yuki

if you can differentiate $\int\limits 1/x dx$ then you will know how to do it. do you need more help ?

3. anonymous

Yes I do, sorry.

4. Yuki

your integral looks awfully like 1/x, it's just shifted a little. you you can guess that it is the family of 1/x. can you integrate 1/x ?

5. anonymous

log x +c?

6. Yuki

it's actually lnx + C

7. anonymous

That's what I meant, lol

8. Yuki

okay just making sure

9. anonymous

I just really don't understand definite integrals.

10. Yuki

now your goal is to make the integral look like that by letting u = 2x+1

11. Yuki

oops, never mind about lnx, i did not see the ^1/2 part. so it will be a family of $\int\limits 1/\sqrt(x)$

12. anonymous

But then what?

13. Yuki

after letting u = 2x-1, $\int\limits 1/\sqrt(u) dx$

14. Yuki

is what you are going to get, but the problem is that the integrand is now a function of u, which cannot be integrated over x. so you have to over come that by figuring out what "dx" is. that technique is the so-called u-substitution. do you know how to find dx ?

15. anonymous

No

16. Yuki

okay, so u = 2x-1 right ? if you differentiate both sides with respect to x, d/dx(u) = d/dx(2x-1) d/dx(u) = 2 do you get this so far ?

17. anonymous

Yes

18. Yuki

so if you multiply dx on both sides and divide 2 on both sides, you will get 1/2 du = dx this is what is happening, but it is traditional and easier to say u=2x-1 du = 2 dx by differentiating both sides

19. anonymous

Gotcha

20. Yuki

so once you substitute dx with the above equation, your integral becomes$1/2\int\limits 1/\sqrt(u) du$

21. Yuki

now it is important to know that the limits of the integration will change as well, unless you want to know how to do that I won't go into the detail. to avoid having to change the limits, you will find out the indefinite integral first, then substitute 2x-1 back into u. that way you can plug in the limits again.

22. anonymous

23. Yuki

$1/2\int\limits 1/\sqrt(u) du = 1/2 *1/2*(u^{1/2})+C$

24. Yuki

1/4(2x-1) + C =F(x) so F(13) -F(5) is your answer. just as a reminder, C will not matter so you can ignore it

25. Yuki

woops$1/4 \sqrt(2x-1) +C$ is what I meant

26. Yuki

it seems like the answer will be 3/4

27. anonymous

Oh, ok, I see. Thanks!!! Can you help me with a couple word problems too?

28. Yuki

I need to go soon, so I can help you with one of them. ask me the one that you think need most help

29. anonymous

A manufacturer has been selling flashlights at $6 a piece, and at this price consumers have been buying 3000 flashlights per month. The manufacturer wishes to raise the price and estimates that for each$1 increase in price, 1000 fewer flashlights will be sold each month. The manufacturer can produce the flashlight a cost of \$4 per flashlight. At what price should the manufacturer sell the flashlights to generate the greatest profit?

30. Yuki

okay so what do you not get from this problem ?