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AllQuestion and Answers: Page 1223

Question Number 85653 Answers: 0 Comments: 1

if f≥0 and (d/dx)(f(x))^2 =(f′(x))^2 and f(0)=1 find f(x) if f(x)=((4x^3 )/(x^2 +1)) find f^( −1) (x) and (f^( −1) )^′ (2)

$${if}\:\:{f}\geqslant\mathrm{0}\:\:{and}\:\:\:\frac{{d}}{{dx}}\left({f}\left({x}\right)\right)^{\mathrm{2}} =\left({f}'\left({x}\right)\right)^{\mathrm{2}} \:{and}\:{f}\left(\mathrm{0}\right)=\mathrm{1} \\ $$$${find}\:{f}\left({x}\right)\:\:\: \\ $$$$ \\ $$$${if}\:{f}\left({x}\right)=\frac{\mathrm{4}{x}^{\mathrm{3}} }{{x}^{\mathrm{2}} +\mathrm{1}}\:\:{find}\:{f}^{\:−\mathrm{1}} \left({x}\right)\:\:\:{and}\:\left({f}^{\:−\mathrm{1}} \right)^{'} \left(\mathrm{2}\right) \\ $$

Question Number 85649 Answers: 1 Comments: 1

Proove that : ((√(2−(√3)))/2) = (((√6)−(√2))/4)

$$\mathrm{Proove}\:\mathrm{that}\:: \\ $$$$ \\ $$$$\frac{\sqrt{\mathrm{2}−\sqrt{\mathrm{3}}}}{\mathrm{2}}\:=\:\frac{\sqrt{\mathrm{6}}−\sqrt{\mathrm{2}}}{\mathrm{4}} \\ $$

Question Number 85648 Answers: 2 Comments: 0

∫(((x^3 −4))/((x+1)))dx

$$\int\frac{\left(\mathrm{x}^{\mathrm{3}} −\mathrm{4}\right)}{\left(\mathrm{x}+\mathrm{1}\right)}\mathrm{dx} \\ $$

Question Number 85646 Answers: 0 Comments: 0

show that ∫(1/([x(x−1)(x−2)(x−3)...(x−m)]^2 ))dx= =(1/((m!)^2 ))Σ_(n=0) ^m ( ((m),(n) )^2 /(n−x))+(2/((m!)^2 ))ln∣Π_(n=0) ^m (x−n)^( ((m),(n) )^2 (H_(m−n) −H_n )) ∣+c

$${show}\:{that} \\ $$$$\int\frac{\mathrm{1}}{\left[{x}\left({x}−\mathrm{1}\right)\left({x}−\mathrm{2}\right)\left({x}−\mathrm{3}\right)...\left({x}−{m}\right)\right]^{\mathrm{2}} }{dx}= \\ $$$$=\frac{\mathrm{1}}{\left({m}!\right)^{\mathrm{2}} }\underset{{n}=\mathrm{0}} {\overset{{m}} {\sum}}\frac{\begin{pmatrix}{{m}}\\{{n}}\end{pmatrix}^{\mathrm{2}} }{{n}−{x}}+\frac{\mathrm{2}}{\left({m}!\right)^{\mathrm{2}} }{ln}\mid\underset{{n}=\mathrm{0}} {\overset{{m}} {\prod}}\left({x}−{n}\right)^{\begin{pmatrix}{{m}}\\{{n}}\end{pmatrix}^{\mathrm{2}} \left({H}_{{m}−{n}} −{H}_{{n}} \right)} \mid+{c} \\ $$

Question Number 85641 Answers: 0 Comments: 2

calculate A_λ =∫_3 ^∞ (dx/((x+λ)^3 (x−2)^4 )) (λ>0)

$${calculate}\:{A}_{\lambda} =\int_{\mathrm{3}} ^{\infty} \:\:\frac{{dx}}{\left({x}+\lambda\right)^{\mathrm{3}} \left({x}−\mathrm{2}\right)^{\mathrm{4}} }\:\:\:\left(\lambda>\mathrm{0}\right) \\ $$

Question Number 85637 Answers: 1 Comments: 2

∫ (dx/(x+(√(x^2 +1))))

$$\int\:\frac{\mathrm{dx}}{\mathrm{x}+\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{1}}}\: \\ $$

Question Number 85630 Answers: 0 Comments: 1

∫_0 ^(2Π) (dx/((√2)−cosx))

$$\int_{\mathrm{0}} ^{\mathrm{2}\Pi} \:\:\:\frac{{dx}}{\sqrt{\mathrm{2}}−{cosx}}\:\: \\ $$

Question Number 85625 Answers: 1 Comments: 0

lim_(x→0) ((x^n −(sin x)^n )/((sin x)^(n+2) ))

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{x}^{\mathrm{n}} −\left(\mathrm{sin}\:\mathrm{x}\right)^{\mathrm{n}} }{\left(\mathrm{sin}\:\mathrm{x}\right)^{\mathrm{n}+\mathrm{2}} } \\ $$

Question Number 85624 Answers: 0 Comments: 0

Question Number 85623 Answers: 1 Comments: 0

prove that sin x cos 2x = (1/(4sin 4x sec x))

$$\mathrm{prove}\:\mathrm{that}\: \\ $$$$\mathrm{sin}\:\mathrm{x}\:\mathrm{cos}\:\mathrm{2x}\:=\:\frac{\mathrm{1}}{\mathrm{4sin}\:\mathrm{4x}\:\mathrm{sec}\:\mathrm{x}} \\ $$

Question Number 85620 Answers: 1 Comments: 0

prove that cosh (x−y)=cosh xcosh y−sinh xsinh y

$${prove}\:{that} \\ $$$$ \\ $$$$\mathrm{cosh}\:\left({x}−{y}\right)=\mathrm{cosh}\:{x}\mathrm{cosh}\:{y}−\mathrm{sinh}\:{x}\mathrm{sinh}\:{y} \\ $$

Question Number 85606 Answers: 0 Comments: 7

Question Number 85603 Answers: 0 Comments: 0

prove the relation ∫_0 ^1 ((li_5 ((x)^(1/5) ))/(x)^(1/5) )dx=(5/4)(((25)/(3072))−((ζ(2))/2^6 )+((ζ(3))/2^4 )−((ζ(4))/2^2 )+ζ(5))

$${prove}\:{the}\:{relation} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{li}_{\mathrm{5}} \left(\sqrt[{\mathrm{5}}]{{x}}\right)}{\sqrt[{\mathrm{5}}]{{x}}}{dx}=\frac{\mathrm{5}}{\mathrm{4}}\left(\frac{\mathrm{25}}{\mathrm{3072}}−\frac{\zeta\left(\mathrm{2}\right)}{\mathrm{2}^{\mathrm{6}} }+\frac{\zeta\left(\mathrm{3}\right)}{\mathrm{2}^{\mathrm{4}} }−\frac{\zeta\left(\mathrm{4}\right)}{\mathrm{2}^{\mathrm{2}} }+\zeta\left(\mathrm{5}\right)\right) \\ $$

Question Number 85592 Answers: 1 Comments: 0

∫(((u+1)^2 )/(u^3 +u))du

$$\int\frac{\left(\mathrm{u}+\mathrm{1}\right)^{\mathrm{2}} }{\mathrm{u}^{\mathrm{3}} +\mathrm{u}}\mathrm{du} \\ $$

Question Number 85591 Answers: 1 Comments: 0

∫((1+4u)/(−4u^2 +2u+2))du

$$\int\frac{\mathrm{1}+\mathrm{4u}}{−\mathrm{4u}^{\mathrm{2}} +\mathrm{2u}+\mathrm{2}}\mathrm{du} \\ $$$$ \\ $$

Question Number 85590 Answers: 0 Comments: 0

∫((1+4u)/(−4u^2 +2u+2))du

$$\int\frac{\mathrm{1}+\mathrm{4u}}{−\mathrm{4u}^{\mathrm{2}} +\mathrm{2u}+\mathrm{2}}\mathrm{du} \\ $$$$ \\ $$

Question Number 85588 Answers: 0 Comments: 6

Question Number 85601 Answers: 0 Comments: 2

∫((4u)/(4u^2 −4u+1))du

$$\int\frac{\mathrm{4u}}{\mathrm{4u}^{\mathrm{2}} −\mathrm{4u}+\mathrm{1}}\mathrm{du} \\ $$

Question Number 85600 Answers: 1 Comments: 3

∫(x^2 /(√(1+x^2 ))) dx

$$\int\frac{{x}^{\mathrm{2}} }{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:{dx} \\ $$

Question Number 85596 Answers: 1 Comments: 1

∫(((√(x+1))−1)/((√(x−1))+1)) dx

$$\int\frac{\sqrt{{x}+\mathrm{1}}−\mathrm{1}}{\sqrt{{x}−\mathrm{1}}+\mathrm{1}}\:{dx} \\ $$

Question Number 85583 Answers: 0 Comments: 0

Question Number 85580 Answers: 0 Comments: 0

Solve: (D^2 +2D+1)y= x cos x

$$\boldsymbol{\mathrm{Solve}}: \\ $$$$\:\left(\mathrm{D}^{\mathrm{2}} +\mathrm{2D}+\mathrm{1}\right)\mathrm{y}=\:\mathrm{x}\:\mathrm{cos}\:\mathrm{x} \\ $$$$ \\ $$

Question Number 85582 Answers: 0 Comments: 1

cos ((π/9))+cos (((2π)/9))+cos (((4π)/9))=

$$\mathrm{cos}\:\left(\frac{\pi}{\mathrm{9}}\right)+\mathrm{cos}\:\left(\frac{\mathrm{2}\pi}{\mathrm{9}}\right)+\mathrm{cos}\:\left(\frac{\mathrm{4}\pi}{\mathrm{9}}\right)= \\ $$

Question Number 85568 Answers: 4 Comments: 2

∫ _0 ^(2π) (dx/((√2)−cos x))

$$\int\underset{\mathrm{0}} {\overset{\mathrm{2}\pi} {\:}}\:\frac{\mathrm{dx}}{\sqrt{\mathrm{2}}−\mathrm{cos}\:\mathrm{x}} \\ $$

Question Number 85557 Answers: 0 Comments: 3

x = (√(1+ (√(5+ (√(11+ (√(19+...)))))))) x = ?

$${x}\:\:=\:\:\sqrt{\mathrm{1}+\:\sqrt{\mathrm{5}+\:\sqrt{\mathrm{11}+\:\sqrt{\mathrm{19}+...}}}} \\ $$$${x}\:\:=\:\:\:? \\ $$

Question Number 85555 Answers: 1 Comments: 0

2x^2 +5x+7=0

$$\mathrm{2}{x}^{\mathrm{2}} +\mathrm{5}{x}+\mathrm{7}=\mathrm{0} \\ $$

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