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Question Number 83354 Answers: 0 Comments: 2

Question Number 83352 Answers: 1 Comments: 6

(√(2sin^2 ((x/2))(1−cos (x)))) = −sin (−x)−5cos (x)

$$\sqrt{\mathrm{2sin}\:^{\mathrm{2}} \left(\frac{\mathrm{x}}{\mathrm{2}}\right)\left(\mathrm{1}−\mathrm{cos}\:\left(\mathrm{x}\right)\right)}\:=\:−\mathrm{sin}\:\left(−\mathrm{x}\right)−\mathrm{5cos}\:\left(\mathrm{x}\right) \\ $$

Question Number 83342 Answers: 1 Comments: 0

∫_0 ^1 ∫_0 ^( z) ∫_y ^1 ((z^(n+1) Li_1 (y))/((zx)^2 ))dx dy dz , ∀ n∈z

$$\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\:{z}} \int_{{y}} ^{\mathrm{1}} \frac{{z}^{{n}+\mathrm{1}} {Li}_{\mathrm{1}} \left({y}\right)}{\left({zx}\right)^{\mathrm{2}} }{dx}\:{dy}\:{dz}\:,\:\forall\:{n}\in{z} \\ $$

Question Number 83341 Answers: 1 Comments: 1

Find the maximum and minimum of the expression 𝚺_(i=1) ^n a_i x_i with 𝚺_(i=1) ^n (x_i −b_i )^2 =c^2 , where a_i , b_i and c are constants. (extracted and modified from Q83331)

$${Find}\:{the}\:{maximum}\:{and}\:{minimum} \\ $$$${of}\:{the}\:{expression}\:\underset{\boldsymbol{{i}}=\mathrm{1}} {\overset{\boldsymbol{{n}}} {\boldsymbol{\sum}}{a}}_{\boldsymbol{{i}}} \boldsymbol{{x}}_{\boldsymbol{{i}}} \:{with} \\ $$$$\underset{\boldsymbol{{i}}=\mathrm{1}} {\overset{\boldsymbol{{n}}} {\boldsymbol{\sum}}}\left(\boldsymbol{{x}}_{\boldsymbol{{i}}} −\boldsymbol{{b}}_{\boldsymbol{{i}}} \right)^{\mathrm{2}} =\boldsymbol{{c}}^{\mathrm{2}} ,\:{where}\:\boldsymbol{{a}}_{\boldsymbol{{i}}} ,\:\boldsymbol{{b}}_{\boldsymbol{{i}}} \:{and}\:\boldsymbol{{c}}\:{are} \\ $$$${constants}. \\ $$$$ \\ $$$$\left({extracted}\:{and}\:{modified}\:{from}\:{Q}\mathrm{83331}\right) \\ $$

Question Number 83340 Answers: 0 Comments: 0

by using the lagrange method solve the partial equatio (p−q=(z/(x+y)))

$${by}\:{using}\:{the}\:{lagrange}\:{method}\:{solve}\:{the}\:{partial}\:{equatio}\:\left({p}−{q}=\frac{{z}}{{x}+{y}}\right) \\ $$

Question Number 83338 Answers: 1 Comments: 0

∫ sin (3x) tan (2x) dx ?

$$\int\:\mathrm{sin}\:\left(\mathrm{3x}\right)\:\mathrm{tan}\:\left(\mathrm{2x}\right)\:\mathrm{dx}\:? \\ $$

Question Number 83331 Answers: 1 Comments: 7

Question Number 83329 Answers: 1 Comments: 1

lim_(x→0) (((e^(−2x) −(1+ax)))/(x^2 (1+bx))) = ?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left(\mathrm{e}^{−\mathrm{2x}} −\left(\mathrm{1}+\mathrm{ax}\right)\right)}{\mathrm{x}^{\mathrm{2}} \:\left(\mathrm{1}+\mathrm{bx}\right)}\:=\:? \\ $$

Question Number 83327 Answers: 0 Comments: 2

(1−x^2 )(d^2 y/dx^2 ) −2x(dy/dx) + p(p+1)y = 0 in descending power of x. what is the solution?

$$\left(\mathrm{1}−\mathrm{x}^{\mathrm{2}} \right)\frac{\mathrm{d}^{\mathrm{2}} \mathrm{y}}{\mathrm{dx}^{\mathrm{2}} }\:−\mathrm{2x}\frac{\mathrm{dy}}{\mathrm{dx}}\:+\:\mathrm{p}\left(\mathrm{p}+\mathrm{1}\right)\mathrm{y}\:=\:\mathrm{0}\: \\ $$$$\mathrm{in}\:\mathrm{descending}\:\mathrm{power}\:\mathrm{of}\:\mathrm{x}.\:\mathrm{what}\:\mathrm{is} \\ $$$$\mathrm{the}\:\mathrm{solution}? \\ $$

Question Number 83323 Answers: 1 Comments: 1

lim_(x→∞) (√((3x−2)(x−(√2)))) − x(√3)−(√2) =?

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\sqrt{\left(\mathrm{3x}−\mathrm{2}\right)\left(\mathrm{x}−\sqrt{\mathrm{2}}\right)}\:−\:\mathrm{x}\sqrt{\mathrm{3}}−\sqrt{\mathrm{2}}\:=? \\ $$$$ \\ $$

Question Number 83319 Answers: 1 Comments: 0

if tan β = ((tan α + tan γ)/(1+tan αtan γ)) show that sin 2β = ((sin 2α+sin 2γ)/(1+sin 2αsin 2γ))

$$\mathrm{if}\:\mathrm{tan}\:\beta\:=\:\frac{\mathrm{tan}\:\alpha\:+\:\mathrm{tan}\:\gamma}{\mathrm{1}+\mathrm{tan}\:\alpha\mathrm{tan}\:\gamma} \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{sin}\:\mathrm{2}\beta\:=\:\frac{\mathrm{sin}\:\mathrm{2}\alpha+\mathrm{sin}\:\mathrm{2}\gamma}{\mathrm{1}+\mathrm{sin}\:\mathrm{2}\alpha\mathrm{sin}\:\mathrm{2}\gamma} \\ $$

Question Number 83318 Answers: 0 Comments: 0

Evaluate ∫x^(1/x^n ) dx ,n>0

$${Evaluate} \\ $$$$\int{x}^{\frac{\mathrm{1}}{{x}^{{n}} }} \:{dx}\:,{n}>\mathrm{0} \\ $$

Question Number 83307 Answers: 0 Comments: 4

Question Number 83297 Answers: 1 Comments: 1

Write down a series expansion for ln [((1−2x)/((1+2x)^2 ))] in ascending powers of x up to and including the term in x^4 . if x is small that terms in x^2 and higher powers are negleted show that (((1−2x)/(1+2x)))^(1/(2x)) ≅ (1 + x)e^(−3)

$$\mathrm{Write}\:\mathrm{down}\:\mathrm{a}\:\mathrm{series}\:\mathrm{expansion}\:\mathrm{for}\: \\ $$$$\:\mathrm{ln}\:\left[\frac{\mathrm{1}−\mathrm{2}{x}}{\left(\mathrm{1}+\mathrm{2}{x}\right)^{\mathrm{2}} }\right]\:\mathrm{in}\:\mathrm{ascending}\:\mathrm{powers}\:\mathrm{of}\:\mathrm{x}\: \\ $$$$\mathrm{up}\:\mathrm{to}\:\mathrm{and}\:\mathrm{including}\:\mathrm{the}\:\mathrm{term}\:\mathrm{in}\:{x}^{\mathrm{4}} .\: \\ $$$$\mathrm{if}\:\mathrm{x}\:\mathrm{is}\:\mathrm{small}\:\mathrm{that}\:\mathrm{terms}\:\mathrm{in}\:\mathrm{x}^{\mathrm{2}} \:\mathrm{and}\:\mathrm{higher}\:\mathrm{powers} \\ $$$$\mathrm{are}\:\mathrm{negleted}\:\mathrm{show}\:\mathrm{that}\:\:\:\left(\frac{\mathrm{1}−\mathrm{2}{x}}{\mathrm{1}+\mathrm{2}{x}}\right)^{\frac{\mathrm{1}}{\mathrm{2}{x}}} \:\cong\:\left(\mathrm{1}\:+\:{x}\right){e}^{−\mathrm{3}} \\ $$$$ \\ $$

Question Number 83296 Answers: 0 Comments: 2

Obtain a maclaurin expansion for a) e^(cos x ) b) e^(cos^2 x)

$$\mathrm{Obtain}\:\mathrm{a}\:\mathrm{maclaurin}\:\mathrm{expansion}\:\mathrm{for}\: \\ $$$$\left.\mathrm{a}\left.\right)\:\mathrm{e}^{\mathrm{cos}\:{x}\:} \:\:\:\:\:\:\:\:\mathrm{b}\right)\:\mathrm{e}^{\mathrm{cos}\:^{\mathrm{2}} {x}} \\ $$

Question Number 83295 Answers: 0 Comments: 0

Expand ln (1 + sinh x) as a series in ascending powers of x up to and including the term in x^3 . Hence , show that (1 + sinh x)^(3/x) ≅ e^2 (1 −x + (x^2 /2))

$$\mathrm{Expand}\:\mathrm{ln}\:\left(\mathrm{1}\:+\:\mathrm{sinh}\:{x}\right)\:\mathrm{as}\:\mathrm{a}\:\mathrm{series}\:\mathrm{in} \\ $$$$\mathrm{ascending}\:\mathrm{powers}\:\mathrm{of}\:{x}\:\mathrm{up}\:\mathrm{to}\:\mathrm{and}\:\mathrm{including} \\ $$$$\mathrm{the}\:\mathrm{term}\:\mathrm{in}\:{x}^{\mathrm{3}} \:.\:\mathrm{Hence}\:,\:\mathrm{show}\:\mathrm{that}\: \\ $$$$\:\left(\mathrm{1}\:+\:\mathrm{sinh}\:{x}\right)^{\frac{\mathrm{3}}{{x}}} \:\cong\:{e}^{\mathrm{2}} \left(\mathrm{1}\:−{x}\:+\:\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\right) \\ $$

Question Number 83293 Answers: 0 Comments: 0

Let A (((−2)),((−1)) ) ,B ((1),(3) ) , C (((−10)),(( 5)) ) three given points in the brand (O,I,J) such as OI=OJ and (OI)⊥(OJ) D is a point such as AD=AC+2 and CD=2 Prove correctly that BD=13 .Can you find the coordinate of D?

$${Let}\:\:{A}\begin{pmatrix}{−\mathrm{2}}\\{−\mathrm{1}}\end{pmatrix}\:\:,{B}\begin{pmatrix}{\mathrm{1}}\\{\mathrm{3}}\end{pmatrix}\:,\:{C}\begin{pmatrix}{−\mathrm{10}}\\{\:\mathrm{5}}\end{pmatrix}\:{three}\:{given}\:{points}\:{in}\:{the}\:{brand}\:\left({O},{I},{J}\right)\:{such}\:{as}\:{OI}={OJ}\:{and}\:\left({OI}\right)\bot\left({OJ}\right) \\ $$$$\:{D}\:{is}\:{a}\:{point}\:{such}\:{as}\:{AD}={AC}+\mathrm{2}\:\:{and}\:\:{CD}=\mathrm{2}\: \\ $$$${Prove}\:{correctly}\:{that}\:\:{BD}=\mathrm{13}\:.{Can}\:{you}\:{find}\:{the}\:{coordinate}\:{of}\:{D}? \\ $$

Question Number 83289 Answers: 3 Comments: 0

solve log_((24sinx)) (24cosx)=(3/2)

$$\:\:{solve} \\ $$$$\boldsymbol{{log}}_{\left(\mathrm{24}\boldsymbol{{sinx}}\right)} \left(\mathrm{24}\boldsymbol{{cosx}}\right)=\frac{\mathrm{3}}{\mathrm{2}} \\ $$

Question Number 83288 Answers: 0 Comments: 0