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Question Number 98424    Answers: 1   Comments: 0

calculste A_n =∫_(−(1/2)) ^(1/2) x^n (√((1−x)/(1+x)))dx find nature of the serie Σ A_n

$$\mathrm{calculste}\:\mathrm{A}_{\mathrm{n}} =\int_{−\frac{\mathrm{1}}{\mathrm{2}}} ^{\frac{\mathrm{1}}{\mathrm{2}}} \:\mathrm{x}^{\mathrm{n}} \sqrt{\frac{\mathrm{1}−\mathrm{x}}{\mathrm{1}+\mathrm{x}}}\mathrm{dx} \\ $$$$\mathrm{find}\:\mathrm{nature}\:\mathrm{of}\:\mathrm{the}\:\mathrm{serie}\:\Sigma\:\mathrm{A}_{\mathrm{n}} \\ $$

Question Number 98423    Answers: 1   Comments: 0

calculate ∫_0 ^∞ e^(−x^2 −(1/x^2 )) dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\mathrm{e}^{−\mathrm{x}^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }} \mathrm{dx} \\ $$

Question Number 98382    Answers: 0   Comments: 2

∫ tan x (√(1+tan^4 x)) dx

$$\int\:\mathrm{tan}\:\mathrm{x}\:\sqrt{\mathrm{1}+\mathrm{tan}\:^{\mathrm{4}} \:\mathrm{x}}\:\mathrm{dx}\: \\ $$

Question Number 98380    Answers: 3   Comments: 0

let {u_n } and {v_n } be sequences defined by u_0 = 9, u_(n+1) = (1/2)u_n −3. v_n = u_n + 6. Calculate P_n = Σ_(i=0) ^n V_i in terms of n, the deduce Q_n = Σ_(i=0) ^n u_i using the above expressions find lim_(x→∞) Q_n .

$$\mathrm{let}\:\left\{{u}_{{n}} \right\}\:\mathrm{and}\:\left\{{v}_{{n}} \right\}\:\mathrm{be}\:\mathrm{sequences}\:\mathrm{defined}\:\mathrm{by} \\ $$$$\:{u}_{\mathrm{0}} \:=\:\mathrm{9},\:{u}_{{n}+\mathrm{1}} \:=\:\frac{\mathrm{1}}{\mathrm{2}}{u}_{{n}} −\mathrm{3}. \\ $$$${v}_{{n}} \:=\:{u}_{{n}} \:+\:\mathrm{6}. \\ $$$$\mathrm{Calculate}\:{P}_{{n}} \:=\:\underset{{i}=\mathrm{0}} {\overset{{n}} {\sum}}{V}_{{i}} \:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:{n},\:\mathrm{the}\:\mathrm{deduce}\:{Q}_{{n}} \:=\:\underset{{i}=\mathrm{0}} {\overset{{n}} {\sum}}{u}_{{i}} \\ $$$$\mathrm{using}\:\mathrm{the}\:\mathrm{above}\:\mathrm{expressions}\:\mathrm{find} \\ $$$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{Q}_{{n}} \:. \\ $$

Question Number 98369    Answers: 2   Comments: 0

If A, B are two square matrices such that AB = A and BA = B, then

$$\mathrm{If}\:{A},\:{B}\:\mathrm{are}\:\mathrm{two}\:\mathrm{square}\:\mathrm{matrices}\:\mathrm{such} \\ $$$$\mathrm{that}\:{AB}\:=\:{A}\:\mathrm{and}\:{BA}\:=\:{B},\:\mathrm{then} \\ $$

Question Number 98368    Answers: 1   Comments: 0

If G^→ is the centroid of a △ABC, then GA^(→) +GB^(→) +GC^(→) =

$$\mathrm{If}\:\overset{\rightarrow} {{G}}\:\mathrm{is}\:\mathrm{the}\:\mathrm{centroid}\:\mathrm{of}\:\mathrm{a}\:\bigtriangleup{ABC},\:\mathrm{then} \\ $$$$\overset{\rightarrow} {{GA}}+\overset{\rightarrow} {{GB}}+\overset{\rightarrow} {{GC}}\:= \\ $$

Question Number 98367    Answers: 1   Comments: 2

show that ϕ(x)=x^2 −x^(−1) is an explicit solution to linear equation (d^2 y/dx^2 ) − ((2y)/x^2 ) = 0

$$\mathrm{show}\:\mathrm{that}\:\varphi\left(\mathrm{x}\right)=\mathrm{x}^{\mathrm{2}} −\mathrm{x}^{−\mathrm{1}} \:\mathrm{is}\:\mathrm{an}\:\mathrm{explicit}\: \\ $$$$\mathrm{solution}\:\mathrm{to}\:\mathrm{linear}\:\mathrm{equation}\:\frac{\mathrm{d}^{\mathrm{2}} \mathrm{y}}{\mathrm{dx}^{\mathrm{2}} }\:−\:\frac{\mathrm{2y}}{\mathrm{x}^{\mathrm{2}} }\:=\:\mathrm{0} \\ $$

Question Number 98355    Answers: 1   Comments: 1

Question Number 98342    Answers: 1   Comments: 2

Question Number 98338    Answers: 2   Comments: 0

∫cos(x^(18) ) dx

$$\int{cos}\left({x}^{\mathrm{18}} \right)\:{dx} \\ $$$$ \\ $$

Question Number 98325    Answers: 1   Comments: 0

Find the curvature vector and its magnitude at any point r^→ = (θ) of the curve r^→ = (acos θ,asin θ,aθ) .Show the locus of the feet of the ⊥ from the origin to the tangent is a curve that completely lies on the hyperbolic x^2 +y^2 −z^2 = a^2

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{curvature}\:\mathrm{vector}\:\mathrm{and} \\ $$$$\mathrm{its}\:\mathrm{magnitude}\:\mathrm{at}\:\mathrm{any}\:\mathrm{point}\: \\ $$$$\overset{\rightarrow} {\mathrm{r}}\:=\:\left(\theta\right)\:\mathrm{of}\:\mathrm{the}\:\mathrm{curve}\:\overset{\rightarrow} {\mathrm{r}}=\:\left(\mathrm{acos}\:\theta,\mathrm{asin}\:\theta,\mathrm{a}\theta\right) \\ $$$$.\mathrm{Show}\:\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:\mathrm{the}\:\mathrm{feet}\:\mathrm{of}\:\mathrm{the} \\ $$$$\bot\:\mathrm{from}\:\mathrm{the}\:\mathrm{origin}\:\mathrm{to}\:\mathrm{the}\:\mathrm{tangent}\: \\ $$$$\mathrm{is}\:\mathrm{a}\:\mathrm{curve}\:\mathrm{that}\:\mathrm{completely}\:\mathrm{lies} \\ $$$$\mathrm{on}\:\mathrm{the}\:\mathrm{hyperbolic}\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} −\mathrm{z}^{\mathrm{2}} =\:\mathrm{a}^{\mathrm{2}} \\ $$

Question Number 98320    Answers: 1   Comments: 0

Question Number 98311    Answers: 1   Comments: 0

calculate ∫_0 ^∞ ((lnx)/((1+x)^2 ))dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{lnx}}{\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{2}} }\mathrm{dx}\: \\ $$

Question Number 98310    Answers: 0   Comments: 0

prove by using serie that ∫_0 ^∞ cos(x^2 )dx =∫_0 ^∞ sin(x^2 )dx

$$\mathrm{prove}\:\mathrm{by}\:\mathrm{using}\:\mathrm{serie}\:\mathrm{that}\:\int_{\mathrm{0}} ^{\infty} \:\mathrm{cos}\left(\mathrm{x}^{\mathrm{2}} \right)\mathrm{dx}\:=\int_{\mathrm{0}} ^{\infty} \mathrm{sin}\left(\mathrm{x}^{\mathrm{2}} \right)\mathrm{dx} \\ $$

Question Number 98309    Answers: 1   Comments: 0

let f(x) = ln( 1 + cosh 2x) show that lim_(x→∞) f(x) = 2x − ln 2 hence deduce lim_(x→−∞) f(x) with the asympotes of the curve.

$$\mathrm{let}\:{f}\left({x}\right)\:=\:\mathrm{ln}\left(\:\mathrm{1}\:+\:\mathrm{cosh}\:\mathrm{2}{x}\right) \\ $$$$\mathrm{show}\:\mathrm{that}\:\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{f}\left({x}\right)\:=\:\mathrm{2}{x}\:−\:\mathrm{ln}\:\mathrm{2} \\ $$$$\mathrm{hence}\:\mathrm{deduce}\:\underset{{x}\rightarrow−\infty} {\mathrm{lim}}\:{f}\left({x}\right)\:\mathrm{with}\:\mathrm{the}\:\mathrm{asympotes}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{curve}. \\ $$

Question Number 98306    Answers: 2   Comments: 1

Question Number 98305    Answers: 2   Comments: 0

calculate ∫_0 ^∞ ((ln(x))/((1+x)^3 ))dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{ln}\left(\mathrm{x}\right)}{\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{3}} }\mathrm{dx} \\ $$

Question Number 98280    Answers: 1   Comments: 0

Let {a_n } be a sequence such that a_1 = 2, a_(n + 1) = ((3a_n + 4)/(2a_n + 3)), n ≥ 1, find a_n

$$\boldsymbol{\mathrm{Let}}\:\:\left\{\boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{n}}} \right\}\:\:\boldsymbol{\mathrm{be}}\:\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{sequence}}\:\boldsymbol{\mathrm{such}}\:\boldsymbol{\mathrm{that}}\:\:\:\boldsymbol{\mathrm{a}}_{\mathrm{1}} \:=\:\:\mathrm{2}, \\ $$$$\boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{n}}\:\:+\:\:\mathrm{1}} \:\:=\:\:\frac{\mathrm{3}\boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{n}}} \:+\:\:\mathrm{4}}{\mathrm{2}\boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{n}}} \:\:+\:\:\mathrm{3}},\:\:\:\:\:\boldsymbol{\mathrm{n}}\:\geqslant\:\mathrm{1},\:\:\:\:\:\boldsymbol{\mathrm{find}}\:\:\:\:\boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{n}}} \\ $$

Question Number 98278    Answers: 1   Comments: 0

Question Number 98275    Answers: 1   Comments: 1

Question Number 98270    Answers: 2   Comments: 0

prove Fg=G((m_1 m_2 )/r^2 )

$${prove} \\ $$$${Fg}={G}\frac{{m}_{\mathrm{1}} {m}_{\mathrm{2}} }{{r}^{\mathrm{2}} } \\ $$

Question Number 98271    Answers: 2   Comments: 5

GivenU_n =∫_0 ^1 x^n (√(1−x))dx n∈N, show that U_n =((2^(n+2) n!(n+1))/((2n+3)!))

$$\mathcal{G}\mathrm{ivenU}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{x}^{\mathrm{n}} \sqrt{\mathrm{1}−\mathrm{x}}\mathrm{dx}\:\:\mathrm{n}\in\mathbb{N},\:\mathrm{show}\:\mathrm{that} \\ $$$$\mathrm{U}_{\mathrm{n}} =\frac{\mathrm{2}^{\mathrm{n}+\mathrm{2}} \mathrm{n}!\left(\mathrm{n}+\mathrm{1}\right)}{\left(\mathrm{2n}+\mathrm{3}\right)!} \\ $$

Question Number 98268    Answers: 0   Comments: 7

let p(x) be a polynomial function of (n−1)^(th) degree and p(k)=k for k=1,2,3,...,n find p(0) and p(n+1). example: n=10

$${let}\:{p}\left({x}\right)\:{be}\:{a}\:{polynomial}\:{function}\:{of} \\ $$$$\left({n}−\mathrm{1}\right)^{{th}} \:{degree}\:{and} \\ $$$${p}\left({k}\right)={k}\:{for}\:{k}=\mathrm{1},\mathrm{2},\mathrm{3},...,{n} \\ $$$${find}\:{p}\left(\mathrm{0}\right)\:{and}\:{p}\left({n}+\mathrm{1}\right). \\ $$$${example}:\:{n}=\mathrm{10} \\ $$

Question Number 98267    Answers: 2   Comments: 0

∀ a,b>0 , a^2 +b^2 =1 prove that ((1/a)+(1/b))((b/(a^2 +1))+(a/(b^2 +1)))≥(8/3)

$$\forall\:{a},{b}>\mathrm{0}\:,\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} =\mathrm{1} \\ $$$${prove}\:{that} \\ $$$$\left(\frac{\mathrm{1}}{{a}}+\frac{\mathrm{1}}{{b}}\right)\left(\frac{{b}}{{a}^{\mathrm{2}} +\mathrm{1}}+\frac{{a}}{{b}^{\mathrm{2}} +\mathrm{1}}\right)\geqslant\frac{\mathrm{8}}{\mathrm{3}} \\ $$

Question Number 98259    Answers: 2   Comments: 4

Question Number 98256    Answers: 2   Comments: 0

∫_0 ^∞ ((log(x))/((√x)(x+1)^2 ))dx

$$\int_{\mathrm{0}} ^{\infty} \frac{{log}\left({x}\right)}{\sqrt{{x}}\left({x}+\mathrm{1}\right)^{\mathrm{2}} }{dx} \\ $$

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