## anonymous 5 years ago Anyone want to help with the integral of x/(x+2)^1/2 from -1 to 2?

1. anonymous

I just don't know how to set it up, what should u be?

2. anonymous

say u = 1/(sqrt(x-2) then you have integral of u/u^2-2. use partial fractions to solve.

3. anonymous

sorry,u = 1/sqrt(x+2)

4. anonymous

lemme know how it works out.

5. anonymous

ok thx I'll try it out

6. anonymous

did you get du=-1/2(x+2)^3/2? or did I do something wrong

7. anonymous

thats right, just checked it

8. anonymous

ok great! . i'd actually not done it. I just looked at the problem and did some calculation in my head. sorry.

9. anonymous

np, wait though not done yet

10. anonymous

so next du=-1(x+2)/2(x+2)^1/2 right?

11. anonymous

umm, you can solve it entirely in u and plug in the values of u, since this is a definite integral.

12. anonymous

no need to deal with x at all

13. anonymous

wait how come? I thought we had to make du look like the remaining terms in the integral, sorry I've just learned integrals today

14. anonymous

Okay. let me get back to you with this. I will take out my pen and paper and write it down. Do ask someone else too.

15. anonymous

ok

16. anonymous

Ok, found it

17. anonymous

hey!

18. anonymous

Ok, so first Let $$u = (x+2)^{1/2} \implies du = (1/2)(x+2)^{-1/2}dx$$ $\implies dx = 2(x+2)^{1/2} du$

19. anonymous

So then $\int_{-1}^2\frac{x}{(x+2)^{1/2}}dx = \int_1^2[u^2-2]du$

20. anonymous

Whoops forgot the 2

21. anonymous

wow, ok let me try that out thanks, do u know of any rules to find these? or to you just have to guess the u sometimes?

22. anonymous

and does the integral change from -1 to 2, to 1 to 2, or is that a typo?

23. anonymous

The integral changes because $$u = (x+2)^{1/2}$$ so when x = -1, u = 1 and when x = 2, u = 2

24. anonymous

right right, forgot about that, when you said you forgot the 2, is it ]2u^2-2]

25. anonymous

$$2[u^2 - 2]$$

26. anonymous

ah right, that would make more sense, well thx again

27. anonymous

Do you see how I arrived at $$2[u^2-2]$$?

28. anonymous

Cause I kinda skipped some steps there.

29. anonymous

well no, but I understand that it's the right equation to the integral

30. anonymous

I can show the steps.

31. anonymous

there are steps to it?

32. anonymous

Yes certainly.

33. anonymous

awesome

34. anonymous

Since $$u = (x+2)^{1/2} \implies x = u^2 -2$$ $$\implies dx = 2u\ du$$ $\implies \frac{x}{(x+2)^{1/2}}dx = \frac{u^2-2}{u} *2u\ du$ $=2[u^2-2]\ du$

35. anonymous

awesome thanks a lot, I don't think we even learned that yet

36. anonymous

do you always implicitly differentiate x? (x'=dx and not 1?)

37. anonymous

I found that out nvm