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Question Number 84313 Answers: 1 Comments: 3

∫ (a^(2/5) −x^(2/5) )^(5/2) dx

$$\int\:\:\left({a}^{\mathrm{2}/\mathrm{5}} −{x}^{\mathrm{2}/\mathrm{5}} \right)^{\frac{\mathrm{5}}{\mathrm{2}}} \:{dx} \\ $$

Question Number 84310 Answers: 0 Comments: 0

Question Number 84309 Answers: 0 Comments: 0

lim_(x→∞) ((1−(ln ∣x∣)^(n+1) )/(1−ln∣x∣))

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{1}−\left(\mathrm{ln}\:\mid\mathrm{x}\mid\right)^{\mathrm{n}+\mathrm{1}} }{\mathrm{1}−\mathrm{ln}\mid\mathrm{x}\mid} \\ $$

Question Number 84304 Answers: 1 Comments: 1

ΔABC AB=a BC=b AC=c and ∠C=90° r=?

$$\Delta\mathrm{ABC}\:\mathrm{AB}=\mathrm{a}\:\mathrm{BC}=\mathrm{b}\:\mathrm{AC}=\mathrm{c}\: \\ $$$$\mathrm{and}\:\angle\mathrm{C}=\mathrm{90}° \\ $$$$\mathrm{r}=? \\ $$

Question Number 84296 Answers: 0 Comments: 1

Question Number 84293 Answers: 1 Comments: 0

solve in R x^([x]) +x^(2−[x]) =x^2 +1

$${solve}\:{in}\:{R} \\ $$$${x}^{\left[{x}\right]} +{x}^{\mathrm{2}−\left[{x}\right]} ={x}^{\mathrm{2}} +\mathrm{1} \\ $$

Question Number 84288 Answers: 1 Comments: 8

Question Number 84283 Answers: 1 Comments: 0

lim_(x→a) (((∣x∣−1)^2 −(∣a∣−1)^2 )/(x−a)) = k , a>0 lim_(x→a) (((∣x∣−1)^3 −(∣a∣−1)^3 )/(x^2 −a^2 )) =

$$\underset{{x}\rightarrow\mathrm{a}} {\mathrm{lim}}\:\frac{\left(\mid\mathrm{x}\mid−\mathrm{1}\right)^{\mathrm{2}} −\left(\mid\mathrm{a}\mid−\mathrm{1}\right)^{\mathrm{2}} }{\mathrm{x}−\mathrm{a}}\:=\:\mathrm{k}\:,\:\mathrm{a}>\mathrm{0} \\ $$$$\underset{{x}\rightarrow\mathrm{a}} {\mathrm{lim}}\:\frac{\left(\mid\mathrm{x}\mid−\mathrm{1}\right)^{\mathrm{3}} −\left(\mid\mathrm{a}\mid−\mathrm{1}\right)^{\mathrm{3}} }{\mathrm{x}^{\mathrm{2}} −\mathrm{a}^{\mathrm{2}} }\:=\: \\ $$$$ \\ $$

Question Number 84275 Answers: 1 Comments: 2

(a.c)b−(a.b)c = a×(c×b) ?

$$\left(\boldsymbol{\mathrm{a}}.\boldsymbol{\mathrm{c}}\right)\boldsymbol{\mathrm{b}}−\left(\boldsymbol{\mathrm{a}}.\boldsymbol{\mathrm{b}}\right)\boldsymbol{\mathrm{c}}\:=\:\boldsymbol{\mathrm{a}}×\left(\boldsymbol{\mathrm{c}}×\boldsymbol{\mathrm{b}}\right)\:? \\ $$

Question Number 84246 Answers: 1 Comments: 0

∫cos^n mx dx =

$$\int\mathrm{cos}^{{n}} \:{mx}\:\:{dx}\:= \\ $$

Question Number 84245 Answers: 0 Comments: 0

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Question Number 84242 Answers: 1 Comments: 1

∫_0 ^(ln2) (1/(cosh(x + ln4)))dx

$$\underset{\mathrm{0}} {\overset{\mathrm{ln2}} {\int}}\frac{\mathrm{1}}{\mathrm{cosh}\left({x}\:+\:\mathrm{ln4}\right)}{dx} \\ $$

Question Number 84236 Answers: 1 Comments: 0

If ax^2 +2hxy +by^2 = 1, show that (d^2 y/dx^2 ) = ((h^2 −ab)/((hx+by)^3 )).

$$\:\:\mathrm{If}\:\boldsymbol{\mathrm{ax}}^{\mathrm{2}} +\mathrm{2}\boldsymbol{\mathrm{hxy}}\:+\boldsymbol{\mathrm{by}}^{\mathrm{2}} =\:\mathrm{1},\:\mathrm{show}\:\mathrm{that} \\ $$$$\:\:\:\:\frac{\boldsymbol{\mathrm{d}}^{\mathrm{2}} \boldsymbol{\mathrm{y}}}{\boldsymbol{\mathrm{dx}}^{\mathrm{2}} }\:=\:\frac{\boldsymbol{\mathrm{h}}^{\mathrm{2}} −\boldsymbol{\mathrm{ab}}}{\left(\boldsymbol{\mathrm{hx}}+\boldsymbol{\mathrm{by}}\right)^{\mathrm{3}} }. \\ $$

Question Number 84234 Answers: 3 Comments: 2

Integrate the following: 1. ∫(√((a+x)/x)) dx 2.∫ (dx/(sin x(3+2 cos x)))

$$\:\:\boldsymbol{\mathrm{Integrate}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{following}}: \\ $$$$\:\:\:\mathrm{1}.\:\int\sqrt{\frac{\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{x}}}{\boldsymbol{\mathrm{x}}}}\:\boldsymbol{\mathrm{dx}} \\ $$$$\:\:\:\mathrm{2}.\int\:\:\frac{\boldsymbol{\mathrm{dx}}}{\boldsymbol{\mathrm{sin}}\:\boldsymbol{\mathrm{x}}\left(\mathrm{3}+\mathrm{2}\:\boldsymbol{\mathrm{cos}}\:\boldsymbol{\mathrm{x}}\right)} \\ $$$$\: \\ $$

Question Number 84263 Answers: 1 Comments: 0

Using the approximation h((dy/dx))_n ≈ y_(n+1) −y_n and that (dy/dx) = 1, y =2 when x = 0 . then , y_1 = [A] h−2 [B] h + 2 [C] h−1 [D] h + 1

$$\mathrm{Using}\:\mathrm{the}\:\mathrm{approximation} \\ $$$$\:{h}\left(\frac{{dy}}{{dx}}\right)_{{n}} \:\approx\:{y}_{{n}+\mathrm{1}} −{y}_{{n}} \:\mathrm{and}\:\mathrm{that}\:\frac{{dy}}{{dx}}\:=\:\mathrm{1},\:{y}\:=\mathrm{2} \\ $$$$\mathrm{when}\:{x}\:=\:\mathrm{0}\:.\:\mathrm{then}\:,\:{y}_{\mathrm{1}} \:= \\ $$$$\left[\mathrm{A}\right]\:{h}−\mathrm{2} \\ $$$$\left[\mathrm{B}\right]\:{h}\:+\:\mathrm{2} \\ $$$$\left[\mathrm{C}\right]\:{h}−\mathrm{1} \\ $$$$\left[\mathrm{D}\right]\:{h}\:+\:\mathrm{1} \\ $$

Question Number 84261 Answers: 3 Comments: 0

Question Number 84259 Answers: 1 Comments: 1

Question Number 84232 Answers: 1 Comments: 2

Solve the following differential equation. (d^2 y/dx^2 ) + (x/(1−x^2 )) (dy/dx)− (y/(1−x^2 ))= x(√(1−x^2 ))

Question Number 84231 Answers: 1 Comments: 0

A compound pendulum oscillates though a small angle θ about its equilibrium position such that 10a((dθ/dt))^2 = 4g cos θ , a >0 . its period is [A] 2π(√(((5a)/(4g)) )) [B] ((2π)/5)(√(a/g)) [C] 2π(√(((2g)/(5a)) )) [D] 2π(√((5a)/g))

$$\mathrm{A}\:\mathrm{compound}\:\mathrm{pendulum}\:\mathrm{oscillates}\:\mathrm{though}\:\mathrm{a} \\ $$$$\mathrm{small}\:\mathrm{angle}\:\theta\:\mathrm{about}\:\mathrm{its}\:\mathrm{equilibrium}\:\mathrm{position} \\ $$$$\mathrm{such}\:\mathrm{that} \\ $$$$\:\:\:\mathrm{10}{a}\left(\frac{{d}\theta}{{dt}}\right)^{\mathrm{2}} \:=\:\mathrm{4}{g}\:\mathrm{cos}\:\theta\:,\:{a}\:>\mathrm{0}\:.\:\mathrm{its}\:\mathrm{period}\:\mathrm{is}\: \\ $$$$\left[\mathrm{A}\right]\:\mathrm{2}\pi\sqrt{\frac{\mathrm{5}{a}}{\mathrm{4}{g}}\:\:}\:\:\:\left[\mathrm{B}\right]\:\frac{\mathrm{2}\pi}{\mathrm{5}}\sqrt{\frac{{a}}{{g}}}\:\:\left[\mathrm{C}\right]\:\mathrm{2}\pi\sqrt{\frac{\mathrm{2}{g}}{\mathrm{5}{a}}\:}\:\:\left[\mathrm{D}\right]\:\mathrm{2}\pi\sqrt{\frac{\mathrm{5}{a}}{{g}}}\: \\ $$

Question Number 84220 Answers: 1 Comments: 1

find the maximum value of (2/(3cosh2x +2))

$$\mathrm{find}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{value}\:\mathrm{of}\:\frac{\mathrm{2}}{\mathrm{3cosh2}{x}\:+\mathrm{2}} \\ $$

Question Number 84219 Answers: 1 Comments: 0

∫_(−1) ^1 e^(∣x∣) dx =?

$$\int_{−\mathrm{1}} ^{\mathrm{1}} {e}^{\mid{x}\mid} {dx}\:=? \\ $$

Question Number 84218 Answers: 4 Comments: 1

find the distance between the planes 2x−y−z = 24 and 2x−y−z = 36

$$\mathrm{find}\:\mathrm{the}\:\mathrm{distance}\:\mathrm{between}\:\mathrm{the}\:\mathrm{planes} \\ $$$$\:\mathrm{2}\boldsymbol{{x}}−\boldsymbol{{y}}−\boldsymbol{{z}}\:=\:\mathrm{24}\:\mathrm{and}\:\mathrm{2}\boldsymbol{{x}}−\boldsymbol{{y}}−\boldsymbol{{z}}\:=\:\mathrm{36} \\ $$

Question Number 84215 Answers: 1 Comments: 0

hi show that the following sequence is limited: U_n =((3n+2)/(2n+1)) precise the upper and lower.

$${hi} \\ $$$${show}\:{that}\:{the}\:{following}\:{sequence} \\ $$$${is}\:{limited}: \\ $$$${U}_{{n}} =\frac{\mathrm{3}{n}+\mathrm{2}}{\mathrm{2}{n}+\mathrm{1}} \\ $$$${precise}\:{the}\:{upper}\:{and}\:{lower}. \\ $$

Question Number 84206 Answers: 0 Comments: 0

a, b, c, d ∈ N (a, b, c, d) is quadruple of a, b, c, d such that b = a^2 + 1 c = b^2 + 1 d = c^2 + 1 τ(a) + τ(b) + τ(c) + τ(d) is odd number(s) . τ(k) is the number of positive divisor of natural number k . a, b, c, d < 10^6 How many quadruple of a, b, c, d ?

$${a},\:{b},\:{c},\:{d}\:\:\in\:\:\mathbb{N} \\ $$$$\left({a},\:{b},\:{c},\:{d}\right)\:\:{is}\:\:{quadruple}\:\:{of}\:\:{a},\:{b},\:{c},\:{d}\:\:{such}\:\:{that} \\ $$$$\:\:\:\:\:\:\:\:{b}\:\:=\:\:{a}^{\mathrm{2}} \:+\:\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:{c}\:\:=\:\:{b}^{\mathrm{2}} \:+\:\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:{d}\:\:=\:\:{c}^{\mathrm{2}} \:+\:\mathrm{1} \\ $$$$\tau\left({a}\right)\:+\:\tau\left({b}\right)\:+\:\tau\left({c}\right)\:+\:\tau\left({d}\right)\:\:{is}\:\:\:{odd}\:\:{number}\left({s}\right)\:. \\ $$$$\tau\left({k}\right)\:\:{is}\:\:{the}\:\:{number}\:\:{of}\:\:{positive}\:\:{divisor}\:\:{of}\:\:\:{natural}\:\:{number}\:\:{k}\:. \\ $$$${a},\:{b},\:{c},\:{d}\:<\:\:\mathrm{10}^{\mathrm{6}} \\ $$$${How}\:\:{many}\:\:{quadruple}\:\:{of}\:\:{a},\:{b},\:{c},\:{d}\:\:? \\ $$

Question Number 84205 Answers: 3 Comments: 0

prove a. sin2x=tanx(1+cos2x) b. sin2x=((2tanx)/(1+tan^2 x))

$${prove}\: \\ $$$${a}.\:{sin}\mathrm{2}{x}={tanx}\left(\mathrm{1}+{cos}\mathrm{2}{x}\right) \\ $$$${b}.\:{sin}\mathrm{2}{x}=\frac{\mathrm{2}{tanx}}{\mathrm{1}+{tan}^{\mathrm{2}} {x}} \\ $$

Question Number 84202 Answers: 1 Comments: 0

prove cos^4 x−sin^4 x=cos22x

$${prove}\: \\ $$$${cos}^{\mathrm{4}} {x}−{sin}^{\mathrm{4}} {x}={cos}\mathrm{22}{x} \\ $$

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