How would you prove $int^8_0frac1{sqrt{x+frac1{sqrt{x}}}}dx<4-frac1{2019}$?












12














We would like to prove the following inequality.




$$int^{8}_{0}frac{1}{sqrt{x+frac{1}{sqrt{x}}}},dx<4-frac{1}{2019}tag{1}$$




What I've tried is using the AM-GM inequality, $$x+frac{1}{sqrt{x}}ge 2,sqrt{x timesfrac{1}{sqrt{x}}}=2x^{frac14} $$ then $$int^{8}_{0}frac{1}{sqrt{x+frac{1}{sqrt{x}}}},dxlefrac{1}{sqrt{2}}int^{8}_{0}frac{1}{sqrt[8]{x}}, dx=4.98cdots<color{red}{5}-frac{1}{2019}.tag{2}$$How would you prove $(1)$?



$color{white}{11110811197115115116117100101110116}$










share|cite|improve this question




















  • 2




    This looks like a contest task; if this is actually the case I would recommend to delete the question hence posting of active contest questions is not allowed here on MSE.
    – mrtaurho
    Dec 14 '18 at 11:23






  • 5




    @mrtaurho No contest task here, just our passionate teacher motivating his students!
    – Urbanmaths
    Dec 14 '18 at 13:34






  • 1




    If you apply AM-GM just on the denominator $sqrt{x cdot 1}$ of the fraction $frac{1}{sqrt{x}}$, you will have that $$I leq int_{0}^{8}sqrt{frac{x+1}{x^2+x+2}} approx 4.07, $$ which is a better result than yours. I think $int_{0}^{8}sqrt{frac{x+1}{x^2+x+2}}$ has a closed form. Not sure!
    – Alex Silva
    Dec 14 '18 at 13:56








  • 2




    @Urbanmaths What exactly do you mean?
    – mrtaurho
    Dec 18 '18 at 10:12






  • 2




    I mean I'm just hoping some comments don't prevent people from answering the above question. Thanks.
    – Urbanmaths
    Dec 18 '18 at 18:04


















12














We would like to prove the following inequality.




$$int^{8}_{0}frac{1}{sqrt{x+frac{1}{sqrt{x}}}},dx<4-frac{1}{2019}tag{1}$$




What I've tried is using the AM-GM inequality, $$x+frac{1}{sqrt{x}}ge 2,sqrt{x timesfrac{1}{sqrt{x}}}=2x^{frac14} $$ then $$int^{8}_{0}frac{1}{sqrt{x+frac{1}{sqrt{x}}}},dxlefrac{1}{sqrt{2}}int^{8}_{0}frac{1}{sqrt[8]{x}}, dx=4.98cdots<color{red}{5}-frac{1}{2019}.tag{2}$$How would you prove $(1)$?



$color{white}{11110811197115115116117100101110116}$










share|cite|improve this question




















  • 2




    This looks like a contest task; if this is actually the case I would recommend to delete the question hence posting of active contest questions is not allowed here on MSE.
    – mrtaurho
    Dec 14 '18 at 11:23






  • 5




    @mrtaurho No contest task here, just our passionate teacher motivating his students!
    – Urbanmaths
    Dec 14 '18 at 13:34






  • 1




    If you apply AM-GM just on the denominator $sqrt{x cdot 1}$ of the fraction $frac{1}{sqrt{x}}$, you will have that $$I leq int_{0}^{8}sqrt{frac{x+1}{x^2+x+2}} approx 4.07, $$ which is a better result than yours. I think $int_{0}^{8}sqrt{frac{x+1}{x^2+x+2}}$ has a closed form. Not sure!
    – Alex Silva
    Dec 14 '18 at 13:56








  • 2




    @Urbanmaths What exactly do you mean?
    – mrtaurho
    Dec 18 '18 at 10:12






  • 2




    I mean I'm just hoping some comments don't prevent people from answering the above question. Thanks.
    – Urbanmaths
    Dec 18 '18 at 18:04
















12












12








12


8





We would like to prove the following inequality.




$$int^{8}_{0}frac{1}{sqrt{x+frac{1}{sqrt{x}}}},dx<4-frac{1}{2019}tag{1}$$




What I've tried is using the AM-GM inequality, $$x+frac{1}{sqrt{x}}ge 2,sqrt{x timesfrac{1}{sqrt{x}}}=2x^{frac14} $$ then $$int^{8}_{0}frac{1}{sqrt{x+frac{1}{sqrt{x}}}},dxlefrac{1}{sqrt{2}}int^{8}_{0}frac{1}{sqrt[8]{x}}, dx=4.98cdots<color{red}{5}-frac{1}{2019}.tag{2}$$How would you prove $(1)$?



$color{white}{11110811197115115116117100101110116}$










share|cite|improve this question















We would like to prove the following inequality.




$$int^{8}_{0}frac{1}{sqrt{x+frac{1}{sqrt{x}}}},dx<4-frac{1}{2019}tag{1}$$




What I've tried is using the AM-GM inequality, $$x+frac{1}{sqrt{x}}ge 2,sqrt{x timesfrac{1}{sqrt{x}}}=2x^{frac14} $$ then $$int^{8}_{0}frac{1}{sqrt{x+frac{1}{sqrt{x}}}},dxlefrac{1}{sqrt{2}}int^{8}_{0}frac{1}{sqrt[8]{x}}, dx=4.98cdots<color{red}{5}-frac{1}{2019}.tag{2}$$How would you prove $(1)$?



$color{white}{11110811197115115116117100101110116}$







calculus integration inequality definite-integrals nested-radicals






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 2 days ago









Olivier Oloa

108k17176293




108k17176293










asked Dec 14 '18 at 11:11









UrbanmathsUrbanmaths

613




613








  • 2




    This looks like a contest task; if this is actually the case I would recommend to delete the question hence posting of active contest questions is not allowed here on MSE.
    – mrtaurho
    Dec 14 '18 at 11:23






  • 5




    @mrtaurho No contest task here, just our passionate teacher motivating his students!
    – Urbanmaths
    Dec 14 '18 at 13:34






  • 1




    If you apply AM-GM just on the denominator $sqrt{x cdot 1}$ of the fraction $frac{1}{sqrt{x}}$, you will have that $$I leq int_{0}^{8}sqrt{frac{x+1}{x^2+x+2}} approx 4.07, $$ which is a better result than yours. I think $int_{0}^{8}sqrt{frac{x+1}{x^2+x+2}}$ has a closed form. Not sure!
    – Alex Silva
    Dec 14 '18 at 13:56








  • 2




    @Urbanmaths What exactly do you mean?
    – mrtaurho
    Dec 18 '18 at 10:12






  • 2




    I mean I'm just hoping some comments don't prevent people from answering the above question. Thanks.
    – Urbanmaths
    Dec 18 '18 at 18:04
















  • 2




    This looks like a contest task; if this is actually the case I would recommend to delete the question hence posting of active contest questions is not allowed here on MSE.
    – mrtaurho
    Dec 14 '18 at 11:23






  • 5




    @mrtaurho No contest task here, just our passionate teacher motivating his students!
    – Urbanmaths
    Dec 14 '18 at 13:34






  • 1




    If you apply AM-GM just on the denominator $sqrt{x cdot 1}$ of the fraction $frac{1}{sqrt{x}}$, you will have that $$I leq int_{0}^{8}sqrt{frac{x+1}{x^2+x+2}} approx 4.07, $$ which is a better result than yours. I think $int_{0}^{8}sqrt{frac{x+1}{x^2+x+2}}$ has a closed form. Not sure!
    – Alex Silva
    Dec 14 '18 at 13:56








  • 2




    @Urbanmaths What exactly do you mean?
    – mrtaurho
    Dec 18 '18 at 10:12






  • 2




    I mean I'm just hoping some comments don't prevent people from answering the above question. Thanks.
    – Urbanmaths
    Dec 18 '18 at 18:04










2




2




This looks like a contest task; if this is actually the case I would recommend to delete the question hence posting of active contest questions is not allowed here on MSE.
– mrtaurho
Dec 14 '18 at 11:23




This looks like a contest task; if this is actually the case I would recommend to delete the question hence posting of active contest questions is not allowed here on MSE.
– mrtaurho
Dec 14 '18 at 11:23




5




5




@mrtaurho No contest task here, just our passionate teacher motivating his students!
– Urbanmaths
Dec 14 '18 at 13:34




@mrtaurho No contest task here, just our passionate teacher motivating his students!
– Urbanmaths
Dec 14 '18 at 13:34




1




1




If you apply AM-GM just on the denominator $sqrt{x cdot 1}$ of the fraction $frac{1}{sqrt{x}}$, you will have that $$I leq int_{0}^{8}sqrt{frac{x+1}{x^2+x+2}} approx 4.07, $$ which is a better result than yours. I think $int_{0}^{8}sqrt{frac{x+1}{x^2+x+2}}$ has a closed form. Not sure!
– Alex Silva
Dec 14 '18 at 13:56






If you apply AM-GM just on the denominator $sqrt{x cdot 1}$ of the fraction $frac{1}{sqrt{x}}$, you will have that $$I leq int_{0}^{8}sqrt{frac{x+1}{x^2+x+2}} approx 4.07, $$ which is a better result than yours. I think $int_{0}^{8}sqrt{frac{x+1}{x^2+x+2}}$ has a closed form. Not sure!
– Alex Silva
Dec 14 '18 at 13:56






2




2




@Urbanmaths What exactly do you mean?
– mrtaurho
Dec 18 '18 at 10:12




@Urbanmaths What exactly do you mean?
– mrtaurho
Dec 18 '18 at 10:12




2




2




I mean I'm just hoping some comments don't prevent people from answering the above question. Thanks.
– Urbanmaths
Dec 18 '18 at 18:04






I mean I'm just hoping some comments don't prevent people from answering the above question. Thanks.
– Urbanmaths
Dec 18 '18 at 18:04












1 Answer
1






active

oldest

votes


















5














For $x>0$, one may check that
begin{align}
small{sqrt {x+frac{1}{sqrt{x}}}}&=sqrt{x}+frac {1}{2x + frac 1{2sqrt{x} +frac{1}{2x+frac{1}{sqrt{x}+sqrt {x+frac{1}{sqrt{x}}}}}}} tag1
end{align}

then, using
begin{align}
small{sqrt{x}+sqrt {x+frac{1}{sqrt{x}}}>2sqrt{x}}
end{align}

one gets




$$int_1^8frac{dx}{sqrt {x+frac{1}{sqrt{x}}}} <int_1^8frac{dx}{sqrt{x}+frac {1}{2x + frac 1{2sqrt{x} +frac{1}{2x+frac{1}{2sqrt{x}}}}}}. tag2
$$




By the change of variable $u=2sqrt{x}$, $du=frac{dx}{sqrt{x}}$, setting $varphi=frac{1+sqrt{5}}2$, one has
begin{align}
&int_1^8frac{dx}{sqrt{x}+frac {1}{2x + frac 1{2sqrt{x} +frac{1}{2x+frac{1}{2sqrt{x}}}}}}
\\&=int_1^{8}frac{16x^{3}+12x^{3/2}+1}{16x^{3}+20x^{3/2}+5}:frac{dx}{sqrt{x}}
\\&=int_2^{4sqrt{2}}frac{u^6+6u^3+4}{u^6+10u^3+20}:du, qquad u=2^{1/3}5^{1/6}t,
\\&=4sqrt{2}-2-2^{4/3}5^{-1/3}varphi int_{2^{2/3}5^{-1/6}}^{2^{13/6}5^{-1/6}}frac{dt}{t^3+varphi}-frac{2^{4/3}5^{2/3}}{varphi}int_{2^{2/3}5^{-1/6}}^{2^{13/6}5^{-1/6}}frac{dt}{t^3+frac1{varphi}}.
end{align}

Then, using the evaluation
$$
int_0^bfrac{dt}{t^3+c^3}=frac{sqrt{3}pi}{18c^2}+frac{sqrt{3}}{3c^2}arctan left(frac{2b-c}{sqrt{3}c}right)-frac{1}{6c^2}ln left(frac{(2b-c)^2+3c^2}{4(b+c)^2}right),,c>0,,b>0,
$$
one gets




$$
int_1^8frac{dx}{sqrt{x}+frac {1}{2x + frac 1{2sqrt{x} +frac{1}{2x+frac{1}{2sqrt{x}}}}}}=3.3199color{red}{58928}cdots. tag3
$$




In a like manner, for $x>0$, one may check that
begin{align}
small{sqrt {x+frac{1}{sqrt{x}}}}&=frac{1}{x^{1/4}}+frac {1}{frac{2}{x^{5/4}} + frac 1{frac{2}{x^{1/4}} +frac{1}{frac{2}{x^{5/4}}+frac{1}{frac{1}{x^{1/4}}+sqrt {x+frac{1}{sqrt{x}}}}}}} tag4
end{align}

then, using
begin{align}
small{frac{1}{x^{1/4}}+sqrt {x+frac{1}{sqrt{x}}}>frac{2}{x^{1/4}}}
end{align}

one gets




$$int_0^1frac{dx}{sqrt {x+frac{1}{sqrt{x}}}} <int_0^1frac{dx}{frac{1}{x^{1/4}}+frac {1}{frac{2}{x^{5/4}} + frac 1{frac{2}{x^{1/4}} +frac{1}{frac{2}{x^{5/4}}+frac{x^{1/4}}2}}}}. tag5
$$




By the change of variable $u=sqrt[4]{x}$, $dx=4u^3du$, setting $varphi=frac{1+sqrt{5}}2$, one has



begin{align}
&int_0^1frac{dx}{frac{1}{x^{1/4}}+frac {1}{frac{2}{x^{5/4}} + frac 1{frac{2}{x^{1/4}} +frac{1}{frac{2}{x^{5/4}}+frac{x^{1/4}}2}}}}
\\&=int_0^{1}frac{x^{3}+12x^{3/2}+16}{5x^{3}+20x^{3/2}+16}:sqrt[4]{x}:dx
\\&=4int_0^1frac{u^{12}+12u^6+16}{5u^{12}+20u^6+16}:u^4du, qquad u=2^{1/3}5^{-1/12}t,
\\&=frac4{25}+frac{6}{sqrt{5}}+2^{-2/3}5^{-1/3}varphi int_0^{2^{-1/3}5^{1/12}}!!frac{t^4dt}{t^6+varphi}-frac{2^{4/3}5^{-1/3}}{varphi}int_0^{2^{-1/3}5^{1/12}}!!frac{t^4dt}{t^6+frac1{varphi}}.
end{align}



Then, using the evaluation
begin{align}
&6cint_0^bfrac{t^4dt}{t^6+c^6}qquad (c>0,,b>0)
\\&=arctan left(sqrt{3}+frac{2b}{c}right)-arctan left(sqrt{3}-frac{2b}{c}right)+2arctan left(frac{b}{c}right)-frac{sqrt{3}}{2}ln left(frac{b^2+sqrt{3}bc+c^2}{b^2+sqrt{3}bc+c^2}right)
end{align}
one obtains




$$
int_0^1frac{dx}{frac{1}{x^{1/4}}+frac {1}{frac{2}{x^{5/4}} + frac 1{frac{2}{x^{1/4}} +frac{1}{frac{2}{x^{5/4}}+frac{x^{1/4}}2}}}}=0.6739color{red}{21415}cdots. tag6
$$




Finally, combining $(2)$, $(3)$, $(5)$ and $(6)$ yields




$$int_0^8frac{dx}{sqrt {x+frac{1}{sqrt{x}}}} <color{red}{3.99}38803cdots<color{red}{4}-frac1{2019}=color{red}{3.99}950cdots tag7
$$




as desired.






share|cite|improve this answer





















  • +1. Of course it's the man himself.
    – dezdichado
    2 days ago










  • That is ingenious.
    – TheSimpliFire
    2 days ago











Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3039230%2fhow-would-you-prove-int8-0-frac1-sqrtx-frac1-sqrtxdx4-frac12019%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









5














For $x>0$, one may check that
begin{align}
small{sqrt {x+frac{1}{sqrt{x}}}}&=sqrt{x}+frac {1}{2x + frac 1{2sqrt{x} +frac{1}{2x+frac{1}{sqrt{x}+sqrt {x+frac{1}{sqrt{x}}}}}}} tag1
end{align}

then, using
begin{align}
small{sqrt{x}+sqrt {x+frac{1}{sqrt{x}}}>2sqrt{x}}
end{align}

one gets




$$int_1^8frac{dx}{sqrt {x+frac{1}{sqrt{x}}}} <int_1^8frac{dx}{sqrt{x}+frac {1}{2x + frac 1{2sqrt{x} +frac{1}{2x+frac{1}{2sqrt{x}}}}}}. tag2
$$




By the change of variable $u=2sqrt{x}$, $du=frac{dx}{sqrt{x}}$, setting $varphi=frac{1+sqrt{5}}2$, one has
begin{align}
&int_1^8frac{dx}{sqrt{x}+frac {1}{2x + frac 1{2sqrt{x} +frac{1}{2x+frac{1}{2sqrt{x}}}}}}
\\&=int_1^{8}frac{16x^{3}+12x^{3/2}+1}{16x^{3}+20x^{3/2}+5}:frac{dx}{sqrt{x}}
\\&=int_2^{4sqrt{2}}frac{u^6+6u^3+4}{u^6+10u^3+20}:du, qquad u=2^{1/3}5^{1/6}t,
\\&=4sqrt{2}-2-2^{4/3}5^{-1/3}varphi int_{2^{2/3}5^{-1/6}}^{2^{13/6}5^{-1/6}}frac{dt}{t^3+varphi}-frac{2^{4/3}5^{2/3}}{varphi}int_{2^{2/3}5^{-1/6}}^{2^{13/6}5^{-1/6}}frac{dt}{t^3+frac1{varphi}}.
end{align}

Then, using the evaluation
$$
int_0^bfrac{dt}{t^3+c^3}=frac{sqrt{3}pi}{18c^2}+frac{sqrt{3}}{3c^2}arctan left(frac{2b-c}{sqrt{3}c}right)-frac{1}{6c^2}ln left(frac{(2b-c)^2+3c^2}{4(b+c)^2}right),,c>0,,b>0,
$$
one gets




$$
int_1^8frac{dx}{sqrt{x}+frac {1}{2x + frac 1{2sqrt{x} +frac{1}{2x+frac{1}{2sqrt{x}}}}}}=3.3199color{red}{58928}cdots. tag3
$$




In a like manner, for $x>0$, one may check that
begin{align}
small{sqrt {x+frac{1}{sqrt{x}}}}&=frac{1}{x^{1/4}}+frac {1}{frac{2}{x^{5/4}} + frac 1{frac{2}{x^{1/4}} +frac{1}{frac{2}{x^{5/4}}+frac{1}{frac{1}{x^{1/4}}+sqrt {x+frac{1}{sqrt{x}}}}}}} tag4
end{align}

then, using
begin{align}
small{frac{1}{x^{1/4}}+sqrt {x+frac{1}{sqrt{x}}}>frac{2}{x^{1/4}}}
end{align}

one gets




$$int_0^1frac{dx}{sqrt {x+frac{1}{sqrt{x}}}} <int_0^1frac{dx}{frac{1}{x^{1/4}}+frac {1}{frac{2}{x^{5/4}} + frac 1{frac{2}{x^{1/4}} +frac{1}{frac{2}{x^{5/4}}+frac{x^{1/4}}2}}}}. tag5
$$




By the change of variable $u=sqrt[4]{x}$, $dx=4u^3du$, setting $varphi=frac{1+sqrt{5}}2$, one has



begin{align}
&int_0^1frac{dx}{frac{1}{x^{1/4}}+frac {1}{frac{2}{x^{5/4}} + frac 1{frac{2}{x^{1/4}} +frac{1}{frac{2}{x^{5/4}}+frac{x^{1/4}}2}}}}
\\&=int_0^{1}frac{x^{3}+12x^{3/2}+16}{5x^{3}+20x^{3/2}+16}:sqrt[4]{x}:dx
\\&=4int_0^1frac{u^{12}+12u^6+16}{5u^{12}+20u^6+16}:u^4du, qquad u=2^{1/3}5^{-1/12}t,
\\&=frac4{25}+frac{6}{sqrt{5}}+2^{-2/3}5^{-1/3}varphi int_0^{2^{-1/3}5^{1/12}}!!frac{t^4dt}{t^6+varphi}-frac{2^{4/3}5^{-1/3}}{varphi}int_0^{2^{-1/3}5^{1/12}}!!frac{t^4dt}{t^6+frac1{varphi}}.
end{align}



Then, using the evaluation
begin{align}
&6cint_0^bfrac{t^4dt}{t^6+c^6}qquad (c>0,,b>0)
\\&=arctan left(sqrt{3}+frac{2b}{c}right)-arctan left(sqrt{3}-frac{2b}{c}right)+2arctan left(frac{b}{c}right)-frac{sqrt{3}}{2}ln left(frac{b^2+sqrt{3}bc+c^2}{b^2+sqrt{3}bc+c^2}right)
end{align}
one obtains




$$
int_0^1frac{dx}{frac{1}{x^{1/4}}+frac {1}{frac{2}{x^{5/4}} + frac 1{frac{2}{x^{1/4}} +frac{1}{frac{2}{x^{5/4}}+frac{x^{1/4}}2}}}}=0.6739color{red}{21415}cdots. tag6
$$




Finally, combining $(2)$, $(3)$, $(5)$ and $(6)$ yields




$$int_0^8frac{dx}{sqrt {x+frac{1}{sqrt{x}}}} <color{red}{3.99}38803cdots<color{red}{4}-frac1{2019}=color{red}{3.99}950cdots tag7
$$




as desired.






share|cite|improve this answer





















  • +1. Of course it's the man himself.
    – dezdichado
    2 days ago










  • That is ingenious.
    – TheSimpliFire
    2 days ago
















5














For $x>0$, one may check that
begin{align}
small{sqrt {x+frac{1}{sqrt{x}}}}&=sqrt{x}+frac {1}{2x + frac 1{2sqrt{x} +frac{1}{2x+frac{1}{sqrt{x}+sqrt {x+frac{1}{sqrt{x}}}}}}} tag1
end{align}

then, using
begin{align}
small{sqrt{x}+sqrt {x+frac{1}{sqrt{x}}}>2sqrt{x}}
end{align}

one gets




$$int_1^8frac{dx}{sqrt {x+frac{1}{sqrt{x}}}} <int_1^8frac{dx}{sqrt{x}+frac {1}{2x + frac 1{2sqrt{x} +frac{1}{2x+frac{1}{2sqrt{x}}}}}}. tag2
$$




By the change of variable $u=2sqrt{x}$, $du=frac{dx}{sqrt{x}}$, setting $varphi=frac{1+sqrt{5}}2$, one has
begin{align}
&int_1^8frac{dx}{sqrt{x}+frac {1}{2x + frac 1{2sqrt{x} +frac{1}{2x+frac{1}{2sqrt{x}}}}}}
\\&=int_1^{8}frac{16x^{3}+12x^{3/2}+1}{16x^{3}+20x^{3/2}+5}:frac{dx}{sqrt{x}}
\\&=int_2^{4sqrt{2}}frac{u^6+6u^3+4}{u^6+10u^3+20}:du, qquad u=2^{1/3}5^{1/6}t,
\\&=4sqrt{2}-2-2^{4/3}5^{-1/3}varphi int_{2^{2/3}5^{-1/6}}^{2^{13/6}5^{-1/6}}frac{dt}{t^3+varphi}-frac{2^{4/3}5^{2/3}}{varphi}int_{2^{2/3}5^{-1/6}}^{2^{13/6}5^{-1/6}}frac{dt}{t^3+frac1{varphi}}.
end{align}

Then, using the evaluation
$$
int_0^bfrac{dt}{t^3+c^3}=frac{sqrt{3}pi}{18c^2}+frac{sqrt{3}}{3c^2}arctan left(frac{2b-c}{sqrt{3}c}right)-frac{1}{6c^2}ln left(frac{(2b-c)^2+3c^2}{4(b+c)^2}right),,c>0,,b>0,
$$
one gets




$$
int_1^8frac{dx}{sqrt{x}+frac {1}{2x + frac 1{2sqrt{x} +frac{1}{2x+frac{1}{2sqrt{x}}}}}}=3.3199color{red}{58928}cdots. tag3
$$




In a like manner, for $x>0$, one may check that
begin{align}
small{sqrt {x+frac{1}{sqrt{x}}}}&=frac{1}{x^{1/4}}+frac {1}{frac{2}{x^{5/4}} + frac 1{frac{2}{x^{1/4}} +frac{1}{frac{2}{x^{5/4}}+frac{1}{frac{1}{x^{1/4}}+sqrt {x+frac{1}{sqrt{x}}}}}}} tag4
end{align}

then, using
begin{align}
small{frac{1}{x^{1/4}}+sqrt {x+frac{1}{sqrt{x}}}>frac{2}{x^{1/4}}}
end{align}

one gets




$$int_0^1frac{dx}{sqrt {x+frac{1}{sqrt{x}}}} <int_0^1frac{dx}{frac{1}{x^{1/4}}+frac {1}{frac{2}{x^{5/4}} + frac 1{frac{2}{x^{1/4}} +frac{1}{frac{2}{x^{5/4}}+frac{x^{1/4}}2}}}}. tag5
$$




By the change of variable $u=sqrt[4]{x}$, $dx=4u^3du$, setting $varphi=frac{1+sqrt{5}}2$, one has



begin{align}
&int_0^1frac{dx}{frac{1}{x^{1/4}}+frac {1}{frac{2}{x^{5/4}} + frac 1{frac{2}{x^{1/4}} +frac{1}{frac{2}{x^{5/4}}+frac{x^{1/4}}2}}}}
\\&=int_0^{1}frac{x^{3}+12x^{3/2}+16}{5x^{3}+20x^{3/2}+16}:sqrt[4]{x}:dx
\\&=4int_0^1frac{u^{12}+12u^6+16}{5u^{12}+20u^6+16}:u^4du, qquad u=2^{1/3}5^{-1/12}t,
\\&=frac4{25}+frac{6}{sqrt{5}}+2^{-2/3}5^{-1/3}varphi int_0^{2^{-1/3}5^{1/12}}!!frac{t^4dt}{t^6+varphi}-frac{2^{4/3}5^{-1/3}}{varphi}int_0^{2^{-1/3}5^{1/12}}!!frac{t^4dt}{t^6+frac1{varphi}}.
end{align}



Then, using the evaluation
begin{align}
&6cint_0^bfrac{t^4dt}{t^6+c^6}qquad (c>0,,b>0)
\\&=arctan left(sqrt{3}+frac{2b}{c}right)-arctan left(sqrt{3}-frac{2b}{c}right)+2arctan left(frac{b}{c}right)-frac{sqrt{3}}{2}ln left(frac{b^2+sqrt{3}bc+c^2}{b^2+sqrt{3}bc+c^2}right)
end{align}
one obtains




$$
int_0^1frac{dx}{frac{1}{x^{1/4}}+frac {1}{frac{2}{x^{5/4}} + frac 1{frac{2}{x^{1/4}} +frac{1}{frac{2}{x^{5/4}}+frac{x^{1/4}}2}}}}=0.6739color{red}{21415}cdots. tag6
$$




Finally, combining $(2)$, $(3)$, $(5)$ and $(6)$ yields




$$int_0^8frac{dx}{sqrt {x+frac{1}{sqrt{x}}}} <color{red}{3.99}38803cdots<color{red}{4}-frac1{2019}=color{red}{3.99}950cdots tag7
$$




as desired.






share|cite|improve this answer





















  • +1. Of course it's the man himself.
    – dezdichado
    2 days ago










  • That is ingenious.
    – TheSimpliFire
    2 days ago














5












5








5






For $x>0$, one may check that
begin{align}
small{sqrt {x+frac{1}{sqrt{x}}}}&=sqrt{x}+frac {1}{2x + frac 1{2sqrt{x} +frac{1}{2x+frac{1}{sqrt{x}+sqrt {x+frac{1}{sqrt{x}}}}}}} tag1
end{align}

then, using
begin{align}
small{sqrt{x}+sqrt {x+frac{1}{sqrt{x}}}>2sqrt{x}}
end{align}

one gets




$$int_1^8frac{dx}{sqrt {x+frac{1}{sqrt{x}}}} <int_1^8frac{dx}{sqrt{x}+frac {1}{2x + frac 1{2sqrt{x} +frac{1}{2x+frac{1}{2sqrt{x}}}}}}. tag2
$$




By the change of variable $u=2sqrt{x}$, $du=frac{dx}{sqrt{x}}$, setting $varphi=frac{1+sqrt{5}}2$, one has
begin{align}
&int_1^8frac{dx}{sqrt{x}+frac {1}{2x + frac 1{2sqrt{x} +frac{1}{2x+frac{1}{2sqrt{x}}}}}}
\\&=int_1^{8}frac{16x^{3}+12x^{3/2}+1}{16x^{3}+20x^{3/2}+5}:frac{dx}{sqrt{x}}
\\&=int_2^{4sqrt{2}}frac{u^6+6u^3+4}{u^6+10u^3+20}:du, qquad u=2^{1/3}5^{1/6}t,
\\&=4sqrt{2}-2-2^{4/3}5^{-1/3}varphi int_{2^{2/3}5^{-1/6}}^{2^{13/6}5^{-1/6}}frac{dt}{t^3+varphi}-frac{2^{4/3}5^{2/3}}{varphi}int_{2^{2/3}5^{-1/6}}^{2^{13/6}5^{-1/6}}frac{dt}{t^3+frac1{varphi}}.
end{align}

Then, using the evaluation
$$
int_0^bfrac{dt}{t^3+c^3}=frac{sqrt{3}pi}{18c^2}+frac{sqrt{3}}{3c^2}arctan left(frac{2b-c}{sqrt{3}c}right)-frac{1}{6c^2}ln left(frac{(2b-c)^2+3c^2}{4(b+c)^2}right),,c>0,,b>0,
$$
one gets




$$
int_1^8frac{dx}{sqrt{x}+frac {1}{2x + frac 1{2sqrt{x} +frac{1}{2x+frac{1}{2sqrt{x}}}}}}=3.3199color{red}{58928}cdots. tag3
$$




In a like manner, for $x>0$, one may check that
begin{align}
small{sqrt {x+frac{1}{sqrt{x}}}}&=frac{1}{x^{1/4}}+frac {1}{frac{2}{x^{5/4}} + frac 1{frac{2}{x^{1/4}} +frac{1}{frac{2}{x^{5/4}}+frac{1}{frac{1}{x^{1/4}}+sqrt {x+frac{1}{sqrt{x}}}}}}} tag4
end{align}

then, using
begin{align}
small{frac{1}{x^{1/4}}+sqrt {x+frac{1}{sqrt{x}}}>frac{2}{x^{1/4}}}
end{align}

one gets




$$int_0^1frac{dx}{sqrt {x+frac{1}{sqrt{x}}}} <int_0^1frac{dx}{frac{1}{x^{1/4}}+frac {1}{frac{2}{x^{5/4}} + frac 1{frac{2}{x^{1/4}} +frac{1}{frac{2}{x^{5/4}}+frac{x^{1/4}}2}}}}. tag5
$$




By the change of variable $u=sqrt[4]{x}$, $dx=4u^3du$, setting $varphi=frac{1+sqrt{5}}2$, one has



begin{align}
&int_0^1frac{dx}{frac{1}{x^{1/4}}+frac {1}{frac{2}{x^{5/4}} + frac 1{frac{2}{x^{1/4}} +frac{1}{frac{2}{x^{5/4}}+frac{x^{1/4}}2}}}}
\\&=int_0^{1}frac{x^{3}+12x^{3/2}+16}{5x^{3}+20x^{3/2}+16}:sqrt[4]{x}:dx
\\&=4int_0^1frac{u^{12}+12u^6+16}{5u^{12}+20u^6+16}:u^4du, qquad u=2^{1/3}5^{-1/12}t,
\\&=frac4{25}+frac{6}{sqrt{5}}+2^{-2/3}5^{-1/3}varphi int_0^{2^{-1/3}5^{1/12}}!!frac{t^4dt}{t^6+varphi}-frac{2^{4/3}5^{-1/3}}{varphi}int_0^{2^{-1/3}5^{1/12}}!!frac{t^4dt}{t^6+frac1{varphi}}.
end{align}



Then, using the evaluation
begin{align}
&6cint_0^bfrac{t^4dt}{t^6+c^6}qquad (c>0,,b>0)
\\&=arctan left(sqrt{3}+frac{2b}{c}right)-arctan left(sqrt{3}-frac{2b}{c}right)+2arctan left(frac{b}{c}right)-frac{sqrt{3}}{2}ln left(frac{b^2+sqrt{3}bc+c^2}{b^2+sqrt{3}bc+c^2}right)
end{align}
one obtains




$$
int_0^1frac{dx}{frac{1}{x^{1/4}}+frac {1}{frac{2}{x^{5/4}} + frac 1{frac{2}{x^{1/4}} +frac{1}{frac{2}{x^{5/4}}+frac{x^{1/4}}2}}}}=0.6739color{red}{21415}cdots. tag6
$$




Finally, combining $(2)$, $(3)$, $(5)$ and $(6)$ yields




$$int_0^8frac{dx}{sqrt {x+frac{1}{sqrt{x}}}} <color{red}{3.99}38803cdots<color{red}{4}-frac1{2019}=color{red}{3.99}950cdots tag7
$$




as desired.






share|cite|improve this answer












For $x>0$, one may check that
begin{align}
small{sqrt {x+frac{1}{sqrt{x}}}}&=sqrt{x}+frac {1}{2x + frac 1{2sqrt{x} +frac{1}{2x+frac{1}{sqrt{x}+sqrt {x+frac{1}{sqrt{x}}}}}}} tag1
end{align}

then, using
begin{align}
small{sqrt{x}+sqrt {x+frac{1}{sqrt{x}}}>2sqrt{x}}
end{align}

one gets




$$int_1^8frac{dx}{sqrt {x+frac{1}{sqrt{x}}}} <int_1^8frac{dx}{sqrt{x}+frac {1}{2x + frac 1{2sqrt{x} +frac{1}{2x+frac{1}{2sqrt{x}}}}}}. tag2
$$




By the change of variable $u=2sqrt{x}$, $du=frac{dx}{sqrt{x}}$, setting $varphi=frac{1+sqrt{5}}2$, one has
begin{align}
&int_1^8frac{dx}{sqrt{x}+frac {1}{2x + frac 1{2sqrt{x} +frac{1}{2x+frac{1}{2sqrt{x}}}}}}
\\&=int_1^{8}frac{16x^{3}+12x^{3/2}+1}{16x^{3}+20x^{3/2}+5}:frac{dx}{sqrt{x}}
\\&=int_2^{4sqrt{2}}frac{u^6+6u^3+4}{u^6+10u^3+20}:du, qquad u=2^{1/3}5^{1/6}t,
\\&=4sqrt{2}-2-2^{4/3}5^{-1/3}varphi int_{2^{2/3}5^{-1/6}}^{2^{13/6}5^{-1/6}}frac{dt}{t^3+varphi}-frac{2^{4/3}5^{2/3}}{varphi}int_{2^{2/3}5^{-1/6}}^{2^{13/6}5^{-1/6}}frac{dt}{t^3+frac1{varphi}}.
end{align}

Then, using the evaluation
$$
int_0^bfrac{dt}{t^3+c^3}=frac{sqrt{3}pi}{18c^2}+frac{sqrt{3}}{3c^2}arctan left(frac{2b-c}{sqrt{3}c}right)-frac{1}{6c^2}ln left(frac{(2b-c)^2+3c^2}{4(b+c)^2}right),,c>0,,b>0,
$$
one gets




$$
int_1^8frac{dx}{sqrt{x}+frac {1}{2x + frac 1{2sqrt{x} +frac{1}{2x+frac{1}{2sqrt{x}}}}}}=3.3199color{red}{58928}cdots. tag3
$$




In a like manner, for $x>0$, one may check that
begin{align}
small{sqrt {x+frac{1}{sqrt{x}}}}&=frac{1}{x^{1/4}}+frac {1}{frac{2}{x^{5/4}} + frac 1{frac{2}{x^{1/4}} +frac{1}{frac{2}{x^{5/4}}+frac{1}{frac{1}{x^{1/4}}+sqrt {x+frac{1}{sqrt{x}}}}}}} tag4
end{align}

then, using
begin{align}
small{frac{1}{x^{1/4}}+sqrt {x+frac{1}{sqrt{x}}}>frac{2}{x^{1/4}}}
end{align}

one gets




$$int_0^1frac{dx}{sqrt {x+frac{1}{sqrt{x}}}} <int_0^1frac{dx}{frac{1}{x^{1/4}}+frac {1}{frac{2}{x^{5/4}} + frac 1{frac{2}{x^{1/4}} +frac{1}{frac{2}{x^{5/4}}+frac{x^{1/4}}2}}}}. tag5
$$




By the change of variable $u=sqrt[4]{x}$, $dx=4u^3du$, setting $varphi=frac{1+sqrt{5}}2$, one has



begin{align}
&int_0^1frac{dx}{frac{1}{x^{1/4}}+frac {1}{frac{2}{x^{5/4}} + frac 1{frac{2}{x^{1/4}} +frac{1}{frac{2}{x^{5/4}}+frac{x^{1/4}}2}}}}
\\&=int_0^{1}frac{x^{3}+12x^{3/2}+16}{5x^{3}+20x^{3/2}+16}:sqrt[4]{x}:dx
\\&=4int_0^1frac{u^{12}+12u^6+16}{5u^{12}+20u^6+16}:u^4du, qquad u=2^{1/3}5^{-1/12}t,
\\&=frac4{25}+frac{6}{sqrt{5}}+2^{-2/3}5^{-1/3}varphi int_0^{2^{-1/3}5^{1/12}}!!frac{t^4dt}{t^6+varphi}-frac{2^{4/3}5^{-1/3}}{varphi}int_0^{2^{-1/3}5^{1/12}}!!frac{t^4dt}{t^6+frac1{varphi}}.
end{align}



Then, using the evaluation
begin{align}
&6cint_0^bfrac{t^4dt}{t^6+c^6}qquad (c>0,,b>0)
\\&=arctan left(sqrt{3}+frac{2b}{c}right)-arctan left(sqrt{3}-frac{2b}{c}right)+2arctan left(frac{b}{c}right)-frac{sqrt{3}}{2}ln left(frac{b^2+sqrt{3}bc+c^2}{b^2+sqrt{3}bc+c^2}right)
end{align}
one obtains




$$
int_0^1frac{dx}{frac{1}{x^{1/4}}+frac {1}{frac{2}{x^{5/4}} + frac 1{frac{2}{x^{1/4}} +frac{1}{frac{2}{x^{5/4}}+frac{x^{1/4}}2}}}}=0.6739color{red}{21415}cdots. tag6
$$




Finally, combining $(2)$, $(3)$, $(5)$ and $(6)$ yields




$$int_0^8frac{dx}{sqrt {x+frac{1}{sqrt{x}}}} <color{red}{3.99}38803cdots<color{red}{4}-frac1{2019}=color{red}{3.99}950cdots tag7
$$




as desired.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Jan 4 at 9:52









Olivier OloaOlivier Oloa

108k17176293




108k17176293












  • +1. Of course it's the man himself.
    – dezdichado
    2 days ago










  • That is ingenious.
    – TheSimpliFire
    2 days ago


















  • +1. Of course it's the man himself.
    – dezdichado
    2 days ago










  • That is ingenious.
    – TheSimpliFire
    2 days ago
















+1. Of course it's the man himself.
– dezdichado
2 days ago




+1. Of course it's the man himself.
– dezdichado
2 days ago












That is ingenious.
– TheSimpliFire
2 days ago




That is ingenious.
– TheSimpliFire
2 days ago


















draft saved

draft discarded




















































Thanks for contributing an answer to Mathematics Stack Exchange!


  • Please be sure to answer the question. Provide details and share your research!

But avoid



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


Use MathJax to format equations. MathJax reference.


To learn more, see our tips on writing great answers.





Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


Please pay close attention to the following guidance:


  • Please be sure to answer the question. Provide details and share your research!

But avoid



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


To learn more, see our tips on writing great answers.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3039230%2fhow-would-you-prove-int8-0-frac1-sqrtx-frac1-sqrtxdx4-frac12019%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







Popular posts from this blog

1300-talet

1300-talet

Display a custom attribute below product name in the front-end Magento 1.9.3.8