tsghernan
  • tsghernan
I can not solve this integral
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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anonymous
  • anonymous
cool story bro
tsghernan
  • tsghernan
\[\int\limits_{1}^{9} \frac{ 1 }{ \sqrt{x*\sqrt{1+\sqrt{x}}} } \delta x \] (The integral is between 9 and 1
g152xx
  • g152xx
@TuringTest is here so have no fear ! :)

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TuringTest
  • TuringTest
Why do some people think I am a math demi-god? I have no idea.... at least not yet.
anonymous
  • anonymous
try rewriting this with fractional powers instead of nested sqrts
anonymous
  • anonymous
then do a u-substitution where u = sqrt(x) (i'm just eyeballing this but it seems like this should work)
TuringTest
  • TuringTest
you ought to try some of the things you suggest or it just becomes a flood of ideas rather than an clear approach that will help the asker.
anonymous
  • anonymous
No, substitute u = 1 + x^(1/2) Then du = (1/2)x^(-1/2) dx and then easy to solve from that point.
anonymous
  • anonymous
heh, same thing, just off by 1 (du is still the same)
TuringTest
  • TuringTest
@tcarroll010 yes, I agree :)
anonymous
  • anonymous
@cnknd , it is necessary to isolate that term of "1" into u along with the x^(1/2)
anonymous
  • anonymous
i don't think it's all that difficult to integrate (1+u)^(-1/4) rather than u^(-1/4)
anonymous
  • anonymous
\[2\int\limits_{}^{}\frac{ du }{ \sqrt{u} }\]
anonymous
  • anonymous
\[= 4\sqrt{u}\]Now, just put the expression for u back in.
anonymous
  • anonymous
\[4\sqrt{1 + \sqrt{x}}\]And you're done.
anonymous
  • anonymous
@tcarroll010 you forgot the a 2nd sqrt. it's \[2\int\limits_{2}^{4}\frac{ du }{ \sqrt{\sqrt{u}} }\]
anonymous
  • anonymous
Did not forget. Go back and look at the du equation. sqrt(x) is already in it.
anonymous
  • anonymous
original eqn had 3 sqrts
anonymous
  • anonymous
They get resolved in the du equation. Simply take the derivative of my answer and you'll see.
anonymous
  • anonymous
the first sqrt from the original expression covers both the x and the sqrt(1+sqrt(x)) unless my eyes are deceiving me.
g152xx
  • g152xx
@cnknd - @tcarroll010 is right, take a look again.
anonymous
  • anonymous
Simply rewrite the original equation to\[\frac{ 1 }{ \sqrt{x}\sqrt{1 + \sqrt{x}} }\]and then it will be clearer. Again, just take the derivative of my answer and you'll get the original equation, written either way.
anonymous
  • anonymous
original integrand was: \[\frac{ 1 }{ \sqrt{x \sqrt{1+\sqrt{x}}} }\]
anonymous
  • anonymous
yes, which is the same as my re-write.
asnaseer
  • asnaseer
@cnknd - @tcarroll010 is right - he is making use of this:\[u = 1 + x^{1/2}\]Then\[du = (1/2)x^{-1/2} dx=\frac{dx}{2\sqrt{x}}\]
anonymous
  • anonymous
let \[A = \sqrt{1+\sqrt{x}}\] the original integrand would be: \[\frac{ 1 }{ \sqrt{xA} }\] and your rewrite is: \[\frac{ 1 }{ \sqrt{x}*A }\] im pretty sure those are different
tsghernan
  • tsghernan
I can solve it. Thank you guys
anonymous
  • anonymous
if you want to double check your answer, here's wolframalpha's solution: http://www.wolframalpha.com/input/?i=integrate+%281%2Fsqrt%28x*sqrt%281%2Bsqrt%28x%29%29%29%29
tsghernan
  • tsghernan
tcarroll010 was right. The best way is substitute u = 1 + x^(1/2)
asnaseer
  • asnaseer
oh - sorry @cnknd - I thought you were wondering where the \(\sqrt{x}\) went. you are right - there should be an extra sqrt there.
asnaseer
  • asnaseer
@tsghernan - please take note of what @cnknd was saying as well - there is an extra sqrt to put in
anonymous
  • anonymous
thank you sir moderator :) and yea u = 1 + x^(1/2) is easier than what i suggested
tsghernan
  • tsghernan
Yes. I know it. I verified the result with the mathematica soft. I reached it. Thank you a lot
anonymous
  • anonymous
I do believe @cnknd has a point. It looks like \[2\int\limits_{}^{}\frac{ du }{ \sqrt[4]{u} }\]is the integral to be considered. How does that look to you now, @cnknd?
anonymous
  • anonymous
yea that's what i meant.
anonymous
  • anonymous
and then (8/3)u^(3/4) with the eventual substitution of u in terms of x. Is that what you are getting?
tsghernan
  • tsghernan
You are right ;)
anonymous
  • anonymous
Good catch @cnknd !
anonymous
  • anonymous
So, the answer is derived with a combination of my substitution equation of u = 1 + x^(1/2) (to resolve the other x^(1/2), that part is correct ) and @cnknd 's correction on the exponent on u going from 1/2 to 1/4.
asnaseer
  • asnaseer
nice to see a "team" working towards a solution :)

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