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Question Number 195689 Answers: 0 Comments: 0

∫_1 ^3 f(x)^3 f′(x)dx=∫_1 ^3 (f(x)^3 )df(x)=∫_1 ^3 t^3 dt=(t^4 /4)∫_1 ^3 =((f(x)^4 )/4)∫_1 3

$$\int_{\mathrm{1}} ^{\mathrm{3}} {f}\left({x}\right)^{\mathrm{3}} {f}'\left({x}\right){dx}=\int_{\mathrm{1}} ^{\mathrm{3}} \left({f}\left({x}\right)^{\mathrm{3}} \right){df}\left({x}\right)=\int_{\mathrm{1}} ^{\mathrm{3}} {t}^{\mathrm{3}} {dt}=\frac{{t}^{\mathrm{4}} }{\mathrm{4}}\int_{\mathrm{1}} ^{\mathrm{3}} =\frac{{f}\left({x}\right)^{\mathrm{4}} }{\mathrm{4}}\int_{\mathrm{1}} \mathrm{3} \\ $$

Question Number 195688 Answers: 1 Comments: 0

Question Number 195681 Answers: 2 Comments: 0

Question Number 195628 Answers: 2 Comments: 3

an unsolved old question #190875 a, b, c are real roots of the equation x^3 −7x^2 +4x+1=0. find (1/( (a)^(1/3) ))+(1/( (b)^(1/3) ))+(1/( (c)^(1/3) ))=?

$$\underline{{an}\:{unsolved}\:{old}\:{question}\:#\mathrm{190875}} \\ $$$${a},\:{b},\:{c}\:{are}\:{real}\:{roots}\:{of}\:{the}\:{equation} \\ $$$${x}^{\mathrm{3}} −\mathrm{7}{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{1}=\mathrm{0}. \\ $$$${find}\:\frac{\mathrm{1}}{\:\sqrt[{\mathrm{3}}]{{a}}}+\frac{\mathrm{1}}{\:\sqrt[{\mathrm{3}}]{{b}}}+\frac{\mathrm{1}}{\:\sqrt[{\mathrm{3}}]{{c}}}=? \\ $$

Question Number 195708 Answers: 2 Comments: 0

Prove that : log_(((√a) − b)) ((√a) +b) = −1

$$\boldsymbol{{Prove}}\:\boldsymbol{{that}}\::\:\boldsymbol{{log}}_{\left(\sqrt{\boldsymbol{{a}}}\:−\:\boldsymbol{{b}}\right)} \left(\sqrt{\boldsymbol{{a}}}\:+\boldsymbol{{b}}\right)\:=\:−\mathrm{1} \\ $$

Question Number 195707 Answers: 1 Comments: 0

Question Number 195619 Answers: 2 Comments: 0

$$\:\:\:\cancel{\underline{\underbrace{ }}} \\ $$

Question Number 195618 Answers: 4 Comments: 0

$$\:\:\:\:\cancel{\underline{\underbrace{ }}} \\ $$

Question Number 195612 Answers: 1 Comments: 0

Question Number 195611 Answers: 1 Comments: 0

Question Number 195608 Answers: 1 Comments: 0

solve : ∫_0 ^( π) (sin x)^(cos x) dx

$$ \\ $$$$ \\ $$$$ \\ $$$$\:\:\:{solve}\::\:\:\:\int_{\mathrm{0}} ^{\:\pi} \:\left(\mathrm{sin}\:{x}\right)^{\mathrm{cos}\:{x}} \:{dx} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 195606 Answers: 0 Comments: 1

Question Number 195602 Answers: 1 Comments: 0

solve ∫ (dx/(sin^(10) (x)+cos^(10) (x)))

$${solve}\:\int\:\frac{{dx}}{{sin}^{\mathrm{10}} \left({x}\right)+{cos}^{\mathrm{10}} \left({x}\right)} \\ $$

Question Number 195597 Answers: 0 Comments: 0

Question Number 195578 Answers: 2 Comments: 0

Question Number 195571 Answers: 2 Comments: 0

let f(x+y)+f(x−y)=2f(x)f(y)∧f((1/2))=−1 compute Σ_(k=1) ^(20) [(1/(sin (k)sin (k+f(k))))]

$${let}\:{f}\left({x}+{y}\right)+{f}\left({x}−{y}\right)=\mathrm{2}{f}\left({x}\right){f}\left({y}\right)\wedge{f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)=−\mathrm{1} \\ $$$${compute}\:\underset{{k}=\mathrm{1}} {\overset{\mathrm{20}} {\sum}}\left[\frac{\mathrm{1}}{\mathrm{sin}\:\left({k}\right)\mathrm{sin}\:\left({k}+{f}\left({k}\right)\right)}\right] \\ $$

Question Number 195570 Answers: 1 Comments: 2

Given three Real numbers (x,y,z),such that x^2 +y^2 +z^2 =1 maximize x^4 +y^4 −2z^4 −3(√2)xyz

$$\mathrm{Given}\:\mathrm{three}\:\mathrm{Real}\:\mathrm{numbers}\:\left({x},{y},{z}\right),{such}\:{that} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} =\mathrm{1} \\ $$$${maximize} \\ $$$${x}^{\mathrm{4}} +{y}^{\mathrm{4}} −\mathrm{2}{z}^{\mathrm{4}} −\mathrm{3}\sqrt{\mathrm{2}}{xyz} \\ $$

Question Number 195569 Answers: 0 Comments: 0

a_i ,b_i ,x_i be reals for i=1,2,3,...,n, such that Σ_(i=1) ^n [a_i x_i ]=0. Prove that (Σ_(i=1) ^n [x_i ^2 ])(Σ_(i=1) ^n [a_i ^2 ]Σ_(i=1) ^n [b_i ^2 ]−(Σ_(i=1) ^n [a_i b_i ])^2 )≥(Σ_(i=1) ^n [a_i ^2 ])(Σ_(i=1) ^n [b_i x_i ])^2

$${a}_{{i}} ,{b}_{{i}} ,{x}_{{i}} {be}\:{reals}\:{for}\:{i}=\mathrm{1},\mathrm{2},\mathrm{3},...,{n},\:{such}\:{that} \\ $$$$\sum_{{i}=\mathrm{1}} ^{{n}} \left[{a}_{{i}} {x}_{{i}} \right]=\mathrm{0}.\:{Prove}\:{that} \\ $$$$\left(\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\left[{x}_{{i}} ^{\mathrm{2}} \right]\right)\left(\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\left[{a}_{{i}} ^{\mathrm{2}} \right]\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\left[{b}_{{i}} ^{\mathrm{2}} \right]−\left(\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\left[{a}_{{i}} {b}_{{i}} \right]\right)^{\mathrm{2}} \right)\geqslant\left(\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\left[{a}_{{i}} ^{\mathrm{2}} \right]\right)\left(\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\left[{b}_{{i}} {x}_{{i}} \right]\right)^{\mathrm{2}} \\ $$

Question Number 195592 Answers: 0 Comments: 2

f(x)=((1376)/((x−1)^(ln((2/(4689)))) )) dom f(x)=? answer this

$${f}\left({x}\right)=\frac{\mathrm{1376}}{\left({x}−\mathrm{1}\right)^{{ln}\left(\frac{\mathrm{2}}{\mathrm{4689}}\right)} } \\ $$$${dom}\:{f}\left({x}\right)=? \\ $$$${answer}\:{this} \\ $$

Question Number 195590 Answers: 1 Comments: 0