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Question Number 116846 Answers: 0 Comments: 2

...nice calculus... prove that :: ∫_0 ^( (π/2)) (√(((2^x −1)sin^3 (x))/((2^x +1)(sin^3 (x)+cos^3 (x))))) dx<(π/8) ...m.n.1970...

$$\:\:\:\:\:\:\:\:\:\:\:\:\:...{nice}\:\:{calculus}... \\ $$$$\:\:\:{prove}\:\:{that}\::: \\ $$$$ \\ $$$$\:\: \\ $$$$\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \sqrt{\frac{\left(\mathrm{2}^{{x}} −\mathrm{1}\right){sin}^{\mathrm{3}} \left({x}\right)}{\left(\mathrm{2}^{{x}} +\mathrm{1}\right)\left({sin}^{\mathrm{3}} \left({x}\right)+{cos}^{\mathrm{3}} \left({x}\right)\right)}}\:\:{dx}<\frac{\pi}{\mathrm{8}} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:...{m}.{n}.\mathrm{1970}... \\ $$$$ \\ $$

Question Number 116845 Answers: 1 Comments: 1

Question Number 116844 Answers: 1 Comments: 0

∫ ((8x+sin^(−1) (2x))/( (√(1−4x^2 )))) dx

$$\int\:\frac{\mathrm{8x}+\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{2x}\right)}{\:\sqrt{\mathrm{1}−\mathrm{4x}^{\mathrm{2}} }}\:\mathrm{dx}\: \\ $$

Question Number 116843 Answers: 0 Comments: 1

Question Number 116841 Answers: 0 Comments: 1

Question Number 116832 Answers: 3 Comments: 0

If 19 sin 2x = 37 cos 2x+38 sin^2 x then tan x = __

$$\mathrm{If}\:\mathrm{19}\:\mathrm{sin}\:\mathrm{2x}\:=\:\mathrm{37}\:\mathrm{cos}\:\mathrm{2x}+\mathrm{38}\:\mathrm{sin}\:^{\mathrm{2}} \mathrm{x} \\ $$$$\mathrm{then}\:\mathrm{tan}\:\mathrm{x}\:=\:\_\_ \\ $$

Question Number 116824 Answers: 2 Comments: 0

Given a>b>0 , a&b real number such that a^2 −ab+b^2 =7 and a−ab+b=−1. find the value of a^2 −b^2

$$\mathrm{Given}\:\mathrm{a}>\mathrm{b}>\mathrm{0}\:,\:\mathrm{a\&b}\:\mathrm{real}\:\mathrm{number}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{a}^{\mathrm{2}} −\mathrm{ab}+\mathrm{b}^{\mathrm{2}} =\mathrm{7}\:\mathrm{and}\:\mathrm{a}−\mathrm{ab}+\mathrm{b}=−\mathrm{1}. \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{a}^{\mathrm{2}} −\mathrm{b}^{\mathrm{2}} \\ $$

Question Number 116822 Answers: 3 Comments: 1

what the value of (√i) =?

$$\:\mathrm{what}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\sqrt{{i}}\:=? \\ $$

Question Number 116820 Answers: 0 Comments: 0

prove that lim f(x)=L and lim f(x)=M, then L=M

$${prove}\:{that}\:{lim}\:{f}\left({x}\right)={L}\:{and}\:{lim}\:{f}\left({x}\right)={M}, \\ $$$${then}\:{L}={M} \\ $$

Question Number 116819 Answers: 1 Comments: 0

prove the limit lim_(x−⟩2) (√(2x))=2

$${prove}\:{the}\:{limit} \\ $$$${li}\underset{{x}−\rangle\mathrm{2}} {{m}}\sqrt{\mathrm{2}{x}}=\mathrm{2} \\ $$

Question Number 116872 Answers: 1 Comments: 0

Question Number 116815 Answers: 2 Comments: 0

∫ (dx/((x−2)(x^2 +4))) =?

$$\int\:\frac{\mathrm{dx}}{\left(\mathrm{x}−\mathrm{2}\right)\left(\mathrm{x}^{\mathrm{2}} +\mathrm{4}\right)}\:=? \\ $$

Question Number 116813 Answers: 2 Comments: 0

∫_1 ^(√3) ((√(1+x^2 ))/x^2 ) dx ?

$$\:\:\:\:\:\:\underset{\mathrm{1}} {\overset{\sqrt{\mathrm{3}}} {\int}}\:\frac{\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }}{\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx}\:? \\ $$

Question Number 116803 Answers: 0 Comments: 0

Solve for X(x,y,z), Y(x,y,z), Z(x,y,z) { (((∂Z/∂y)−(∂Y/∂z)=1−x^2 )),(((∂Z/∂x)−(∂X/∂z)=−(y^2 /2))),(((∂Y/∂x)−(∂X/∂y)=z(2x−y))) :} where { ((X(x,y,0)=0)),((Y(x,y,0)=0)),((Z(x,y,z)=0)) :}

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{X}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right),\:\mathrm{Y}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right),\:\mathrm{Z}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right) \\ $$$$\begin{cases}{\frac{\partial\mathrm{Z}}{\partial\mathrm{y}}−\frac{\partial\mathrm{Y}}{\partial\mathrm{z}}=\mathrm{1}−\mathrm{x}^{\mathrm{2}} }\\{\frac{\partial\mathrm{Z}}{\partial\mathrm{x}}−\frac{\partial\mathrm{X}}{\partial\mathrm{z}}=−\frac{\mathrm{y}^{\mathrm{2}} }{\mathrm{2}}}\\{\frac{\partial\mathrm{Y}}{\partial\mathrm{x}}−\frac{\partial\mathrm{X}}{\partial\mathrm{y}}=\mathrm{z}\left(\mathrm{2x}−\mathrm{y}\right)}\end{cases}\:\mathrm{where}\:\begin{cases}{\mathrm{X}\left(\mathrm{x},\mathrm{y},\mathrm{0}\right)=\mathrm{0}}\\{\mathrm{Y}\left(\mathrm{x},\mathrm{y},\mathrm{0}\right)=\mathrm{0}}\\{\mathrm{Z}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)=\mathrm{0}}\end{cases} \\ $$

Question Number 116806 Answers: 2 Comments: 0

Hi Prove that: ∫_(-∞) ^(+∞) -e^(-x^2 ) dx=(√π) Thanks beforehand

$$\mathrm{Hi} \\ $$$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\:\int_{-\infty} ^{+\infty} -\mathrm{e}^{-\mathrm{x}^{\mathrm{2}} } \mathrm{dx}=\sqrt{\pi} \\ $$$$\mathrm{Thanks}\:\mathrm{beforehand} \\ $$$$ \\ $$

Question Number 116800 Answers: 1 Comments: 0

Question Number 116798 Answers: 1 Comments: 0

... calculus... a,b,c ∈R^(+ ) :: find min((√( ((b+c)/a))) +(√((a+c)/b)) +(√((a+b)/c)) )=??? ... m.n.1970...

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\:\:{calculus}... \\ $$$$ \\ $$$$\:\:\:\:\:{a},{b},{c}\:\in\mathbb{R}^{+\:} :: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:{find} \\ $$$$ \\ $$$$\:\:\:\:{min}\left(\sqrt{\:\frac{{b}+{c}}{{a}}}\:+\sqrt{\frac{{a}+{c}}{{b}}}\:+\sqrt{\frac{{a}+{b}}{{c}}}\:\right)=??? \\ $$$$\:\: \\ $$$$\:\:\:\:\:\:\:\:\:...\:{m}.{n}.\mathrm{1970}... \\ $$$$\:\: \\ $$

Question Number 116797 Answers: 0 Comments: 0

Question Number 116796 Answers: 1 Comments: 0

... calculus ... prove that :: ∫_0 ^( 1) (((1−x^p )(1−x^q )x^(r−1) )/(log(x)))dx=^(???) log( (((p+q+r+1)r)/((p+r)(q+r))) ) m.n.1970

$$\:\:\:\:\:\:\:\:...\:\:\:\:\:\:{calculus}\:\:... \\ $$$$ \\ $$$$\:\:\:\:\:{prove}\:\:{that}\::: \\ $$$$ \\ $$$$\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\left(\mathrm{1}−{x}^{{p}} \right)\left(\mathrm{1}−{x}^{{q}} \right){x}^{{r}−\mathrm{1}} }{{log}\left({x}\right)}{dx}\overset{???} {=}{log}\left(\:\frac{\left({p}+{q}+{r}+\mathrm{1}\right){r}}{\left({p}+{r}\right)\left({q}+{r}\right)}\:\right) \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:{m}.{n}.\mathrm{1970} \\ $$$$\: \\ $$

Question Number 116793 Answers: 0 Comments: 0

..calculus.. x,y,z ∈R^+ and x^2 +y^2 +z^2 =1 find min_(x,y,z∈R^(+ ) ) ((((yz)/x)+((xz)/y)+((xy)/z)) )=? m.n.1970..

$$\:\:\:\:\:\:\:..{calculus}.. \\ $$$$\:\:{x},{y},{z}\:\in\mathbb{R}^{+} \:\:{and}\:\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \:=\mathrm{1} \\ $$$$\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{find}\:\:\:\:\:\: \\ $$$$\:\:\:\:{min}_{{x},{y},{z}\in\mathbb{R}^{+\:\:\:\:} } \left(\left(\frac{{yz}}{{x}}+\frac{{xz}}{{y}}+\frac{{xy}}{{z}}\right)\:\right)=? \\ $$$$ \\ $$$$\:\:\:\:\:\:\:{m}.{n}.\mathrm{1970}.. \\ $$

Question Number 116780 Answers: 2 Comments: 1

Question Number 116779 Answers: 1 Comments: 0

solve x^x =2

$${solve} \\ $$$${x}^{{x}} =\mathrm{2} \\ $$

Question Number 116776 Answers: 1 Comments: 0

Given function: f(x)= { ((2x^2 +1, ∣x∣<3)),((5x−1, ∣x∣≥3)) :} Find out: f(x^2 +7)=? A)5x^2 −34 B)2x^2 +8 C)5x^2 +36 D)5x^2 +34 E)2(x^2 +7)^2 +1 Please help

$$\mathrm{Given}\:\mathrm{function}: \\ $$$${f}\left({x}\right)=\begin{cases}{\mathrm{2}{x}^{\mathrm{2}} +\mathrm{1},\:\:\mid{x}\mid<\mathrm{3}}\\{\mathrm{5x}−\mathrm{1},\:\:\mid{x}\mid\geqslant\mathrm{3}}\end{cases} \\ $$$$\mathrm{Find}\:\mathrm{out}:\:{f}\left({x}^{\mathrm{2}} +\mathrm{7}\right)=? \\ $$$$ \\ $$$$\left.\mathrm{A}\left.\right)\left.\mathrm{5x}^{\mathrm{2}} −\mathrm{34}\:\:\:\:\:\mathrm{B}\right)\mathrm{2x}^{\mathrm{2}} +\mathrm{8}\:\:\:\mathrm{C}\right)\mathrm{5x}^{\mathrm{2}} +\mathrm{36} \\ $$$$\left.\mathrm{D}\left.\right)\mathrm{5x}^{\mathrm{2}} +\mathrm{34}\:\:\:\:\mathrm{E}\right)\mathrm{2}\left(\mathrm{x}^{\mathrm{2}} +\mathrm{7}\right)^{\mathrm{2}} +\mathrm{1} \\ $$$$\mathrm{Please}\:\mathrm{help} \\ $$

Question Number 116771 Answers: 0 Comments: 0

Solve the differential equation (dy/dx)−(y/x^2 )=((√(y^2 −1))/x) Please help

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{differential}\:\mathrm{equation} \\ $$$$\frac{\mathrm{dy}}{\mathrm{dx}}−\frac{\mathrm{y}}{\mathrm{x}^{\mathrm{2}} }=\frac{\sqrt{\mathrm{y}^{\mathrm{2}} −\mathrm{1}}}{\mathrm{x}} \\ $$$$\mathrm{Please}\:\mathrm{help} \\ $$

Question Number 116770 Answers: 1 Comments: 1

how to prove the Pythagorean theorem(c^2 =a^2 +b^2 )

$${how}\:{to}\:{prove}\:{the}\:{Pythagorean}\:{theorem}\left({c}^{\mathrm{2}} ={a}^{\mathrm{2}} +{b}^{\mathrm{2}} \right) \\ $$

Question Number 116768 Answers: 2 Comments: 0

sin (4sin^(−1) (x)) = sin (2sin^(−1) (x))

$$\mathrm{sin}\:\left(\mathrm{4sin}^{−\mathrm{1}} \left(\mathrm{x}\right)\right)\:=\:\mathrm{sin}\:\left(\mathrm{2sin}^{−\mathrm{1}} \left(\mathrm{x}\right)\right) \\ $$

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