Question

Anyone want to help with the integral of x/(x+2)^1/2 from -1 to 2?

Answer 1

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

Answer 2

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

Answer 3

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

Answer 4

lemme know how it works out.

Answer 5

ok thx I'll try it out

Answer 6

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

Answer 7

thats right, just checked it

Answer 8

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

Answer 9

np, wait though not done yet

Answer 10

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

Answer 11

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

Answer 12

no need to deal with x at all

Answer 13

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

Answer 14

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.

Answer 15

ok

Answer 16

Ok, found it

Answer 17

hey!

Answer 18

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

Answer 19

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

Answer 20

Whoops forgot the 2

Answer 21

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?

Answer 22

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

Answer 23

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

Answer 24

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

Answer 25

\(2[u^2 - 2]\)

Answer 26

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

Answer 27

Do you see how I arrived at \(2[u^2-2]\)?

Answer 28

Cause I kinda skipped some steps there.

Answer 29

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

Answer 30

I can show the steps.

Answer 31

there are steps to it?

Answer 32

Yes certainly.

Answer 33

awesome

Answer 34

Since
\( u = (x+2)^{1/2} \implies x = u^2 -2\)
\(\implies dx = 2u\ du\)

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

Answer 35

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

Answer 36

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

Answer 37

I found that out nvm