## anonymous 5 years ago How do you take the antiderivative of (2x-1)^-1/2???

1. anonymous

Is that (2x-1)^(-1/2) or (2x-1)^-1 which is divided by 2?

2. anonymous

(2x-1)^(-1/2)

3. anonymous

O.k. Then you would let u = 2x-1 Then du = 2 dx So your integral:$\int\limits \frac{1}{\sqrt{2x+1}}dx$ becomes

4. anonymous

1/2 times $\int\limits \frac{2dx}{\sqrt{2x+1}}$

5. anonymous

$\int\limits \frac{du}{\sqrt{u}}=$

6. anonymous

$\int\limits u ^{-1/2}du=$

7. anonymous

$\frac{u ^{-1/2 + 1}}{-1/2+1}+C$

8. anonymous

$\frac{(2x+1) ^{1/2}}{1/2}+C$

9. anonymous

and dividing by 1/2 is the same as multiplying by the reciprocal 2/1

10. anonymous

What about x/(2x-1)^(1/2) with an upper of 5 and a lower of 1. My calculator is giving me a correcct answer of 5.333333. But I can work the problem out to be the same

11. anonymous

sorry, I "can't" work the problem out the same

12. anonymous

o.k. well, you would want to plug 5 into $2*\sqrt{2x+1}$ and then subtract what you get when you plug 1 into it.

13. anonymous

that would be 2 times the result of $\sqrt{2(5)+1} - \sqrt{2(1)+1} = \sqrt{11}-\sqrt{3}$

14. anonymous

which turns out to be about 3.17

15. anonymous

Oh! I didn't notice you had an "x" in the numerator.

16. anonymous

You would still use the same substitution

17. anonymous

but the x in the numerator can be rewritten in terms of "u"

18. anonymous

right. x= (1+u)/2

19. anonymous

since u = 2x+1 u-1 = 2x and so you get that x = (.5)*(u - 1)

20. anonymous

so dividing that by $\sqrt{u} = u^{1/2}$ you can use exponent rules to get

21. anonymous

$\frac{.5*(u-1)}{u^{1/2}}$

22. anonymous

which boils down to $.5* (u^{1-1/2}-u^{-1/2})$

23. anonymous

which is$.5* (u^{1/2}-u^{-1/2})$

24. anonymous

and you can use the power rule for integrals to antidifferentiate

25. anonymous

$.5* (u^{3/2}-u^{1/2})$

26. anonymous

or $.5* ((2x+1)^{3/2}-(2x+1)^{1/2})$

27. anonymous

does that make sense?

28. anonymous

Yes, but when I plug in the upper and lower I'm still not getting the correct answer.......

29. anonymous

let me check

30. anonymous

The (2x+1) is suppose to be (2x-1)

31. anonymous

right...it's 14.8511...

32. anonymous

Oh! :) Well, I can guarantee you if you use 2x-1 instead it should work but I'll check

33. anonymous

My graphing calculator is giving me 5.333333

34. anonymous

$9^{3/2}-9^{1/2}-(1^{3/2}-1^{1/2})$

35. anonymous

is $3\sqrt{3}-3$

36. anonymous

Do you have a graphing cal on you?

37. anonymous

Actually, you have to go back and change the substitution also so it is more work than this

38. anonymous

if u = 2x-1 then x=.5*(u+1)

39. anonymous

Thanks for your help. I'm in FLorida and it's about 1am. I need to hit the hay. Thanks again......

40. anonymous

Yeah, I did it by hand and it works out. Just use that substitution

41. anonymous

and also multiply by the reciprocal of the new exponent you get when you integrate. ;)

42. anonymous

Will do. Thanks!

43. anonymous

Your welcome...sorry I'm new to typing this out. It's a bit distracting ;) but I'll get used to it. Night!

44. anonymous

Yeah, it's a pain in the retrice Laters