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Question Number 204642 Answers: 1 Comments: 0

If (1/1^2 )+(1/2^2 )+(1/3^2 )+ (1/4^2 )+(1/5^2 ) + ............. = (π^2 /6) then (1/1^2 )+(1/3^2 )+(1/5^2 ) + ............. = ?

$$\mathrm{If}\:\:\frac{\mathrm{1}}{\mathrm{1}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{2}} }+\:\frac{\mathrm{1}}{\mathrm{4}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{5}^{\mathrm{2}} }\:+\:.............\:=\:\frac{\pi^{\mathrm{2}} }{\mathrm{6}} \\ $$$$\mathrm{then}\:\:\frac{\mathrm{1}}{\mathrm{1}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{5}^{\mathrm{2}} }\:+\:.............\:=\:? \\ $$$$ \\ $$

Question Number 204640 Answers: 1 Comments: 0

f(x)=(1/( (√(1+x))))+(1/( (√(1+a))))+(√((ax)/(ax+8))) a>0 x>0 prove 1<f(x)<2

$${f}\left({x}\right)=\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+{x}}}+\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+{a}}}+\sqrt{\frac{{ax}}{{ax}+\mathrm{8}}} \\ $$$${a}>\mathrm{0}\:{x}>\mathrm{0} \\ $$$${prove}\:\mathrm{1}<{f}\left({x}\right)<\mathrm{2} \\ $$

Question Number 204632 Answers: 3 Comments: 0

Question Number 204628 Answers: 2 Comments: 0

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Question Number 204621 Answers: 3 Comments: 0

a , b , c ∈ R^+ If (√a) + (√b) + (√c) = 1 Prove that: a + b + c ≥ (1/3)

$$\mathrm{a}\:,\:\mathrm{b}\:,\:\mathrm{c}\:\in\:\mathbb{R}^{+} \\ $$$$\mathrm{If}\:\:\:\sqrt{\mathrm{a}}\:+\:\sqrt{\mathrm{b}}\:+\:\sqrt{\mathrm{c}}\:=\:\mathrm{1} \\ $$$$\mathrm{Prove}\:\mathrm{that}:\:\:\:\mathrm{a}\:+\:\mathrm{b}\:+\:\mathrm{c}\:\geqslant\:\frac{\mathrm{1}}{\mathrm{3}} \\ $$

Question Number 204618 Answers: 1 Comments: 0

if 7x=(π/2)→((cosxsin2xtan3x)/(cot4xcos5xsin6x))=?

$${if}\:\:\mathrm{7}{x}=\frac{\pi}{\mathrm{2}}\rightarrow\frac{{cosxsin}\mathrm{2}{xtan}\mathrm{3}{x}}{{cot}\mathrm{4}{xcos}\mathrm{5}{xsin}\mathrm{6}{x}}=? \\ $$

Question Number 204617 Answers: 0 Comments: 1

Question Number 204615 Answers: 0 Comments: 2

((exercice )/) prouver ∫_0 ^𝛑 ∫_0 ^x sin((x^2 /𝛑))dxdy=𝛑 ...............prof cedric junior...........

$$\:\:\:\:\:\:\:\:\frac{\boldsymbol{\mathrm{exercice}}\:}{} \\ $$$$\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{prouver}}\:\int_{\mathrm{0}} ^{\boldsymbol{\pi}} \int_{\mathrm{0}} ^{\boldsymbol{\mathrm{x}}} \boldsymbol{{sin}}\left(\frac{\boldsymbol{\mathrm{x}}^{\mathrm{2}} }{\boldsymbol{\pi}}\right)\boldsymbol{{d}\mathrm{x}{d}\mathrm{y}}=\boldsymbol{\pi} \\ $$$$\: \\ $$$$\:\:...............\boldsymbol{{prof}}\:\boldsymbol{{cedric}}\:\boldsymbol{{junior}}........... \\ $$$$ \\ $$

Question Number 204610 Answers: 1 Comments: 0

If , f(x) = { (( 2^(2x) − log_3 ( x+3 ) ; x ≥5)),(( f (1+ x ) −4 ; x < 5)) :} ⇒ f (0 )= ?

$$ \\ $$$$\:\:\:{If}\:,\:\:\:{f}\left({x}\right)\:=\:\begin{cases}{\:\mathrm{2}^{\mathrm{2}{x}} −\:{log}_{\mathrm{3}} \:\left(\:{x}+\mathrm{3}\:\right)\:\:\:\:;\:\:\:{x}\:\geqslant\mathrm{5}}\\{\:{f}\:\left(\mathrm{1}+\:{x}\:\right)\:\:−\mathrm{4}\:\:\:\:\:\:\:\:\:\:\:\:\:\:;\:\:{x}\:<\:\mathrm{5}}\end{cases}\:\:\:\:\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\Rightarrow\:\:{f}\:\left(\mathrm{0}\:\right)=\:? \\ $$$$ \\ $$

Question Number 204603 Answers: 0 Comments: 4

Question Number 204598 Answers: 1 Comments: 2

∫((x−x^2 ))^(1/3) dx

$$\:\:\:\:\:\:\:\:\int\sqrt[{\mathrm{3}}]{\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{x}}^{\mathrm{2}} }\:\boldsymbol{\mathrm{dx}} \\ $$$$ \\ $$

Question Number 204595 Answers: 1 Comments: 0

f(x) = x^3 − 16x^2 − 57x +1 f(a)= 0 f(b)=0 f(c)=0 (a)^(1/5) + ((b ))^(1/5) + ((c ))^(1/5) = ?

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}\right)\:=\:\boldsymbol{\mathrm{x}}^{\mathrm{3}} \:−\:\mathrm{16}\boldsymbol{\mathrm{x}}^{\mathrm{2}} \:−\:\mathrm{57}\boldsymbol{\mathrm{x}}\:+\mathrm{1}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{a}}\right)=\:\mathrm{0}\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{b}}\right)=\mathrm{0}\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{c}}\right)=\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\sqrt[{\mathrm{5}}]{\boldsymbol{\mathrm{a}}}\:+\:\sqrt[{\mathrm{5}}]{\boldsymbol{\mathrm{b}}\:}\:+\:\sqrt[{\mathrm{5}}]{\boldsymbol{\mathrm{c}}\:}\:=\:? \\ $$

Question Number 204590 Answers: 1 Comments: 0

Question Number 204583 Answers: 1 Comments: 0

For z = a − bi If (∣z∣ − z)∙(∣z∣ + z^(−) ) = 4bi Find ∣z∣ = ?

$$\mathrm{For}\:\:\:\mathrm{z}\:=\:\mathrm{a}\:−\:\mathrm{bi} \\ $$$$\mathrm{If}\:\:\:\left(\mid\mathrm{z}\mid\:−\:\mathrm{z}\right)\centerdot\left(\mid\mathrm{z}\mid\:+\:\overline {\mathrm{z}}\right)\:=\:\mathrm{4bi} \\ $$$$\mathrm{Find}\:\:\:\mid\mathrm{z}\mid\:=\:? \\ $$

Question Number 204573 Answers: 0 Comments: 3

How Can derive LambertW(z) in the Form of integral??? W(z)=(1/π)∫_0 ^( π) ln(1+((z∙sin(t))/t)e^(t∙cot(t)) )dt , z∈[−(1/e),∞) Or Similar to the example.LambertW(z) How other Functions can be Derived in Integral Form

$$\mathrm{How}\:\mathrm{Can}\:\mathrm{derive}\:\mathrm{LambertW}\left({z}\right)\:\mathrm{in}\:\mathrm{the} \\ $$$$\:\mathrm{Form}\:\mathrm{of}\:\mathrm{integral}??? \\ $$$$\mathrm{W}\left({z}\right)=\frac{\mathrm{1}}{\pi}\int_{\mathrm{0}} ^{\:\pi} \:\mathrm{ln}\left(\mathrm{1}+\frac{{z}\centerdot\mathrm{sin}\left({t}\right)}{{t}}{e}^{{t}\centerdot\mathrm{cot}\left({t}\right)} \right)\mathrm{d}{t}\:,\:{z}\in\left[−\frac{\mathrm{1}}{{e}},\infty\right) \\ $$$$\mathrm{Or}\:\mathrm{Similar}\:\mathrm{to}\:\mathrm{the}\:\mathrm{example}.\mathrm{LambertW}\left({z}\right) \\ $$$$\mathrm{How}\:\mathrm{other}\:\mathrm{Functions}\:\mathrm{can}\:\mathrm{be}\:\mathrm{Derived}\:\mathrm{in}\:\mathrm{Integral}\:\mathrm{Form} \\ $$

Question Number 204569 Answers: 1 Comments: 0

find the value of I=∫_0 ^(+∞) ln(1+e^(−x) )dx nowing that Σ_(n=1) ^(+∞) (1/n^2 )=(π^2 /6)

$$\boldsymbol{{find}}\:\boldsymbol{{the}}\:\boldsymbol{{value}}\:\boldsymbol{{of}}\: \\ $$$$\boldsymbol{{I}}=\int_{\mathrm{0}} ^{+\infty} \boldsymbol{{ln}}\left(\mathrm{1}+\boldsymbol{{e}}^{−\boldsymbol{{x}}} \right)\boldsymbol{{dx}}\:\boldsymbol{{nowing}}\:\boldsymbol{{that}}\: \\ $$$$\underset{{n}=\mathrm{1}} {\overset{+\infty} {\sum}}\frac{\mathrm{1}}{\boldsymbol{{n}}^{\mathrm{2}} }=\frac{\pi^{\mathrm{2}} }{\mathrm{6}} \\ $$

Question Number 204568 Answers: 1 Comments: 0

how to convert 31230 in base 60? pls help

$$\boldsymbol{{how}}\:\boldsymbol{{to}}\:\boldsymbol{{convert}}\:\mathrm{31230}\:\boldsymbol{{in}}\:\boldsymbol{{base}}\:\mathrm{60}? \\ $$$$\boldsymbol{{pls}}\:\boldsymbol{{help}} \\ $$

Question Number 204574 Answers: 2 Comments: 0

lim_(n→∞) n^(−3/2) [(n+1)^((n+1)) (n+2)^((n+2)) ...(2n)^(2n) ]^(1/n^2 ) = ?

$$ \\ $$$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{n}^{−\mathrm{3}/\mathrm{2}} \left[\left(\mathrm{n}+\mathrm{1}\right)^{\left(\mathrm{n}+\mathrm{1}\right)} \left(\mathrm{n}+\mathrm{2}\right)^{\left(\mathrm{n}+\mathrm{2}\right)} ...\left(\mathrm{2n}\right)^{\mathrm{2n}} \right]^{\mathrm{1}/\mathrm{n}^{\mathrm{2}} } \:=\:? \\ $$$$ \\ $$

Question Number 204560 Answers: 2 Comments: 1

Question Number 204558 Answers: 0 Comments: 0

lim_(n→∞) n^(−3/2) [(n+1)^((n+1)) (n+2)^((n+2)) ...(2n)^(2n) ]^(1/n^2 ) = ?

$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{n}^{−\mathrm{3}/\mathrm{2}} \left[\left(\mathrm{n}+\mathrm{1}\right)^{\left(\mathrm{n}+\mathrm{1}\right)} \left(\mathrm{n}+\mathrm{2}\right)^{\left(\mathrm{n}+\mathrm{2}\right)} ...\left(\mathrm{2n}\right)^{\mathrm{2n}} \right]^{\mathrm{1}/\mathrm{n}^{\mathrm{2}} } \:=\:? \\ $$

Question Number 204545 Answers: 2 Comments: 0

If a = (1/2^2 ) + (1/3^2 ) + ... + (1/(100^2 )) b = 0,99 Prove that: a < b

$$\mathrm{If} \\ $$$$\mathrm{a}\:=\:\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}} }\:\:+\:\:\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{2}} }\:\:+\:\:...\:\:+\:\:\frac{\mathrm{1}}{\mathrm{100}^{\mathrm{2}} } \\ $$$$\mathrm{b}\:=\:\mathrm{0},\mathrm{99} \\ $$$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\mathrm{a}\:<\:\mathrm{b} \\ $$

Question Number 204541 Answers: 0 Comments: 4

In a regular pentagon PQRST , PR intersects QS at O. Calculate ROS?

$${In}\:{a}\:{regular}\:{pentagon}\:{PQRST}\:,\:{PR} \\ $$$${intersects}\:{QS}\:{at}\:{O}.\:{Calculate}\:{ROS}? \\ $$

Question Number 204533 Answers: 1 Comments: 0

Question Number 204522 Answers: 1 Comments: 3

Question Number 204521 Answers: 1 Comments: 1

Question Number 204517 Answers: 2 Comments: 0

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