Question

Show that d/dx [(2x)/√(x+1)] = [(x+2)/(x+1)^(3/2)]. Hence evaluate Intergration from 0 to 8 [(x+2)/(x+1)^(3/2)] dx.

Answer 1

do you know the quotient rule in derivatives ?

Answer 2

Yes

Answer 3

so did you try it ?
d/dx [(2x)/√(x+1)] = ... ?

Answer 4

Nope, I wasn't sure what to do. Hold on, I'll try it and get back to you .Thanks :)

Answer 5

sure, take your time :)

Answer 6

It equals [(x+2)/(x+1)^(3/2)]

Answer 7

so you could successfully do the first part, right ?
how about 2nd part, any ideas ?

Answer 8

So am I supposed to integrate the value I just found?

Answer 9

no,
integration is anti-derivative!

\(\Large if, \dfrac{d}{dx}f(x) = F(x) \\\Large then, \int F(x) = f(x) +c\)

Answer 10

so integration of
[(x+2)/(x+1)^(3/2)] dx
is just
(2x)/√(x+1)
makes sense ?

Answer 11

It's just the reverse right?

Answer 12

yup, thats correct :)

Answer 13

and we first plug in upper limit, then, the lower limit

so plug in x =8 in [(2x)/√(x+1)]

then x =0

Answer 14

Then proceed to integrate it?

Answer 15

what?
you need not integrate anything here

Answer 16

just because derivative of [(2x)/√(x+1)] = [(x+2)/(x+1)^(3/2)]

so integration of [(x+2)/(x+1)^(3/2)] is [(2x)/√(x+1)]

Answer 17

Oh okay, I get it now. I just need to sub in the x values as the limits?

Answer 18

yes, thats right

Answer 19

and then subtract them

Answer 20

Alright thank you :D Appreciate your help

Answer 21

welcome ^_^