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AllQuestion and Answers: Page 1142

Question Number 91133    Answers: 1   Comments: 2

lim_(x→1) (((x−1)+((1−x))^(1/(3 )) )/((1−x^2 ))^(1/(3 )) ) =

$$\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{\left({x}−\mathrm{1}\right)+\sqrt[{\mathrm{3}\:\:}]{\mathrm{1}−{x}}}{\sqrt[{\mathrm{3}\:\:}]{\mathrm{1}−{x}^{\mathrm{2}} }}\:=\: \\ $$

Question Number 91119    Answers: 0   Comments: 0

∫_0 ^( ∫_0 ^( k) (1 + (1/x))^x dx) sin (x^e ) dx = (π/e) k = ?

$$\: \\ $$$$\:\int_{\mathrm{0}} ^{\:\int_{\mathrm{0}} ^{\:{k}} \:\left(\mathrm{1}\:+\:\frac{\mathrm{1}}{{x}}\right)^{{x}} {dx}} \:\mathrm{sin}\:\left({x}^{{e}} \right)\:{dx}\:=\:\frac{\pi}{{e}} \\ $$$$\:{k}\:=\:? \\ $$

Question Number 91099    Answers: 1   Comments: 12

Question Number 91088    Answers: 0   Comments: 0

let U_n =∫_0 ^(1/2) (dx/(√(1−x^n ))) calculate lim_(n→+∞) U_n

$${let}\:{U}_{{n}} =\int_{\mathrm{0}} ^{\frac{\mathrm{1}}{\mathrm{2}}} \:\frac{{dx}}{\sqrt{\mathrm{1}−{x}^{{n}} }}\:\:{calculate}\:{lim}_{{n}\rightarrow+\infty} \:{U}_{{n}} \\ $$

Question Number 91086    Answers: 1   Comments: 0

∫ ((2−x^2 )/(1+x(√(1−x^2 )))) dx

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

Question Number 91085    Answers: 0   Comments: 1

f(x) = ⌈2x−1⌉ f(3) , f(4) = ?

$${f}\left({x}\right)\:=\:\lceil\mathrm{2}{x}−\mathrm{1}\rceil\: \\ $$$${f}\left(\mathrm{3}\right)\:,\:{f}\left(\mathrm{4}\right)\:=\:? \\ $$

Question Number 91074    Answers: 1   Comments: 1

this trig integral has quite a few insights on trig integrals andd u subs as well as on the properties of logarithms. try it out it′s a nice one ∫tan(x)dx

$${this}\:{trig}\:{integral}\:{has}\:{quite}\:{a}\:{few}\: \\ $$$${insights}\:{on}\:{trig}\:{integrals}\:{andd}\: \\ $$$${u}\:{subs}\:{as}\:{well}\:{as}\:{on}\:{the}\:{properties} \\ $$$${of}\:{logarithms}.\:{try}\:{it}\:{out}\:{it}'{s}\:{a}\:{nice} \\ $$$${one} \\ $$$$\int{tan}\left({x}\right){dx} \\ $$

Question Number 91054    Answers: 1   Comments: 2

Question Number 91047    Answers: 1   Comments: 6

Find the square root of: (√7) + (√5)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{square}\:\mathrm{root}\:\mathrm{of}:\:\:\:\:\sqrt{\mathrm{7}}\:\:+\:\:\sqrt{\mathrm{5}} \\ $$

Question Number 91038    Answers: 1   Comments: 1

if sin((α/2))=(4/5) and cos((β/2))=(3/5) prove sin(α)=cos(β)

$${if}\:{sin}\left(\frac{\alpha}{\mathrm{2}}\right)=\frac{\mathrm{4}}{\mathrm{5}} \\ $$$${and}\:{cos}\left(\frac{\beta}{\mathrm{2}}\right)=\frac{\mathrm{3}}{\mathrm{5}} \\ $$$${prove} \\ $$$${sin}\left(\alpha\right)={cos}\left(\beta\right) \\ $$

Question Number 91037    Answers: 0   Comments: 0

hi every one what is the scientific reason for using trigonometric compensation in integration? and what is the rule that we rely on in other compensation? (√(a^2 −x^2 ))→→x=a sin(θ) OR a cos(θ) (√(a^2 +x^2 ))→→x=a tan(θ) OR a cot(θ) (√(x^2 −a^2 ))→→x=a sec(θ) or a csc(θ)

$${hi}\:{every}\:{one} \\ $$$${what}\:{is}\:{the}\:{scientific}\:{reason}\:{for}\:{using} \\ $$$${trigonometric}\:{compensation}\: \\ $$$${in}\:{integration}? \\ $$$${and}\:{what}\:{is}\:{the}\:{rule}\:{that}\:{we}\:{rely}\:{on} \\ $$$${in}\:{other}\:{compensation}? \\ $$$$ \\ $$$$\sqrt{{a}^{\mathrm{2}} −{x}^{\mathrm{2}} }\rightarrow\rightarrow{x}={a}\:{sin}\left(\theta\right)\:{OR}\:{a}\:{cos}\left(\theta\right) \\ $$$$\sqrt{{a}^{\mathrm{2}} +{x}^{\mathrm{2}} }\rightarrow\rightarrow{x}={a}\:{tan}\left(\theta\right)\:{OR}\:{a}\:{cot}\left(\theta\right) \\ $$$$\sqrt{{x}^{\mathrm{2}} −{a}^{\mathrm{2}} }\rightarrow\rightarrow{x}={a}\:{sec}\left(\theta\right)\:{or}\:{a}\:{csc}\left(\theta\right) \\ $$

Question Number 91036    Answers: 1   Comments: 0

Solve: (x+a)^2 (d^2 y/dx^2 )−4(x+a)(dy/dx)+6y= x

$$\:\mathrm{Solve}: \\ $$$$\:\:\left(\mathrm{x}+\mathrm{a}\right)^{\mathrm{2}} \frac{\mathrm{d}^{\mathrm{2}} \mathrm{y}}{\mathrm{dx}^{\mathrm{2}} }−\mathrm{4}\left(\mathrm{x}+\mathrm{a}\right)\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{6y}=\:\mathrm{x} \\ $$

Question Number 91024    Answers: 1   Comments: 5

x^2 y′′+3xy′ +2y = 4x^2

$${x}^{\mathrm{2}} {y}''+\mathrm{3}{xy}'\:+\mathrm{2}{y}\:=\:\mathrm{4}{x}^{\mathrm{2}} \\ $$

Question Number 91020    Answers: 0   Comments: 2

lim_(x→0) (sin x)^(1/x) ?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\mathrm{sin}\:{x}\right)^{\frac{\mathrm{1}}{{x}}} \:?\: \\ $$

Question Number 91018    Answers: 0   Comments: 1

∫_0 ^π ((sin((21x)/2))/(sin(x/2)))dx

$$\int_{\mathrm{0}} ^{\pi} \frac{{sin}\frac{\mathrm{21}{x}}{\mathrm{2}}}{{sin}\frac{{x}}{\mathrm{2}}}{dx} \\ $$

Question Number 91012    Answers: 1   Comments: 0

(dy/dx) + 2xy = xe^(−x^2 ) y^3

$$\frac{{dy}}{{dx}}\:+\:\mathrm{2}{xy}\:=\:{xe}^{−{x}^{\mathrm{2}} } {y}^{\mathrm{3}} \\ $$

Question Number 91011    Answers: 2   Comments: 0

how to make the sigma and prod sign bigger?

$$\:\mathrm{how}\:\mathrm{to}\:\mathrm{make}\:\mathrm{the}\:\mathrm{sigma}\:\mathrm{and}\:\mathrm{prod}\:\mathrm{sign}\:\mathrm{bigger}? \\ $$

Question Number 91010    Answers: 1   Comments: 2

solve the diff eq y′′′−y′′+4y′−4y= e^x

$${solve}\:{the}\:{diff}\:{eq}\: \\ $$$${y}'''−{y}''+\mathrm{4}{y}'−\mathrm{4}{y}=\:{e}^{{x}} \\ $$

Question Number 91000    Answers: 0   Comments: 2

Solve the differential equations: (d^2 y/dx^2 )+ (x/(1−x^2 )) (dy/dx)− (y/(1−x^2 ))= x(√(1−x^2 ))

$$\:\:\boldsymbol{\mathrm{Solve}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{differential}}\:\boldsymbol{\mathrm{equations}}: \\ $$$$\:\:\:\frac{\boldsymbol{\mathrm{d}}^{\mathrm{2}} \boldsymbol{\mathrm{y}}}{\boldsymbol{\mathrm{dx}}^{\mathrm{2}} }+\:\frac{\boldsymbol{\mathrm{x}}}{\mathrm{1}−\boldsymbol{\mathrm{x}}^{\mathrm{2}} }\:\frac{\boldsymbol{\mathrm{dy}}}{\boldsymbol{\mathrm{dx}}}−\:\frac{\boldsymbol{\mathrm{y}}}{\mathrm{1}−\boldsymbol{\mathrm{x}}^{\mathrm{2}} }=\:\boldsymbol{\mathrm{x}}\sqrt{\mathrm{1}−\boldsymbol{\mathrm{x}}^{\mathrm{2}} } \\ $$

Question Number 90997    Answers: 0   Comments: 3

∫ _0 ^π ((sin x dx)/(1+sin x))

$$\int\underset{\mathrm{0}} {\overset{\pi} {\:}}\:\frac{\mathrm{sin}\:{x}\:{dx}}{\mathrm{1}+\mathrm{sin}\:{x}} \\ $$

Question Number 90989    Answers: 1   Comments: 1

Question Number 90983    Answers: 0   Comments: 1

y′′−5y′+6y=x^2

$${y}''−\mathrm{5}{y}'+\mathrm{6}{y}={x}^{\mathrm{2}} \\ $$

Question Number 91068    Answers: 1   Comments: 0

Find the sum (√(1+(1/2^2 )+(1/3^2 )))+(√(1+(1/3^2 )+(1/4^2 )))+...+(√(1+(1/(999^2 ))+(1/(1000^2 ))))

$${Find}\:{the}\:{sum} \\ $$$$\sqrt{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{2}} }}+\sqrt{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{4}^{\mathrm{2}} }}+...+\sqrt{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{999}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{1000}^{\mathrm{2}} }} \\ $$

Question Number 90974    Answers: 0   Comments: 0

solve cost y^(′′) −2sint y^′ +2cost y =e^t

$${solve}\:{cost}\:{y}^{''} \:−\mathrm{2}{sint}\:{y}^{'} \:+\mathrm{2}{cost}\:{y}\:={e}^{{t}} \\ $$

Question Number 90973    Answers: 1   Comments: 2

solve xy^′ +(x+1)y =e^(−x) ln(1+x^2 )

$${solve}\:{xy}^{'} \:+\left({x}+\mathrm{1}\right){y}\:={e}^{−{x}} {ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right) \\ $$

Question Number 90972    Answers: 1   Comments: 0

solve y^(′′) +y =(2/(sin^2 t))

$${solve}\:{y}^{''} \:+{y}\:=\frac{\mathrm{2}}{{sin}^{\mathrm{2}} {t}} \\ $$

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