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AllQuestion and Answers: Page 994

Question Number 120060 Answers: 2 Comments: 0

(i) ∫_(−2) ^0 (dx/(2x+3)) (ii)∫_3 ^5 (dx/( (((4−x)^2 ))^(1/3) ))

$$\left({i}\right)\:\underset{−\mathrm{2}} {\overset{\mathrm{0}} {\int}}\:\frac{{dx}}{\mathrm{2}{x}+\mathrm{3}} \\ $$$$\left({ii}\right)\underset{\mathrm{3}} {\overset{\mathrm{5}} {\int}}\:\frac{{dx}}{\:\sqrt[{\mathrm{3}}]{\left(\mathrm{4}−{x}\right)^{\mathrm{2}} }}\: \\ $$

Question Number 120059 Answers: 1 Comments: 0

Question Number 120058 Answers: 1 Comments: 0

(d^2 y/dx) +x (dy/dx) −y=0

$$\:\frac{{d}^{\mathrm{2}} {y}}{{dx}}\:+{x}\:\frac{{dy}}{{dx}}\:−{y}=\mathrm{0} \\ $$

Question Number 120050 Answers: 3 Comments: 0

(i) y′′−4y′+5y=4sin^2 4x (ii) (x/2)+1 = (√(∣1−x^2 ∣))

$$\:\left({i}\right)\:{y}''−\mathrm{4}{y}'+\mathrm{5}{y}=\mathrm{4sin}\:^{\mathrm{2}} \mathrm{4}{x} \\ $$$$\:\left({ii}\right)\:\frac{{x}}{\mathrm{2}}+\mathrm{1}\:=\:\sqrt{\mid\mathrm{1}−{x}^{\mathrm{2}} \mid}\: \\ $$

Question Number 120049 Answers: 2 Comments: 0

f(x+2)+f(x−2)=f(x) f(1)=1 ,f(2)=2,f(3)=3,f(4)=4 then f(100)=?

$$\:{f}\left({x}+\mathrm{2}\right)+{f}\left({x}−\mathrm{2}\right)={f}\left({x}\right) \\ $$$${f}\left(\mathrm{1}\right)=\mathrm{1}\:,{f}\left(\mathrm{2}\right)=\mathrm{2},{f}\left(\mathrm{3}\right)=\mathrm{3},{f}\left(\mathrm{4}\right)=\mathrm{4} \\ $$$${then}\:{f}\left(\mathrm{100}\right)=? \\ $$

Question Number 120044 Answers: 2 Comments: 0

Given a_(n+1) = ((2a_n )/((2n+1)(2n+2))) find a_n .

$${Given}\:{a}_{{n}+\mathrm{1}} \:=\:\frac{\mathrm{2}{a}_{{n}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)\left(\mathrm{2}{n}+\mathrm{2}\right)} \\ $$$${find}\:{a}_{{n}} . \\ $$

Question Number 120040 Answers: 2 Comments: 0

∫ (t^5 /( (√(2+t^2 )))) dt

$$\:\int\:\frac{{t}^{\mathrm{5}} }{\:\sqrt{\mathrm{2}+{t}^{\mathrm{2}} }}\:{dt}\: \\ $$

Question Number 120037 Answers: 1 Comments: 0

Suppose that R>0, x_0 >0, and x_(n+1) =(1/2)((R/x_n )+x_n ), n≥0 Prove: For n≥1, x_n >x_(n+1) >(√R) and x_n −(√R)≤(1/2^n ) (((x_0 −(√R))^2 )/x_0 )

$$\mathrm{Suppose}\:\mathrm{that}\:\mathrm{R}>\mathrm{0},\:\mathrm{x}_{\mathrm{0}} >\mathrm{0},\:\mathrm{and} \\ $$$$\mathrm{x}_{\mathrm{n}+\mathrm{1}} =\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\mathrm{R}}{\mathrm{x}_{\mathrm{n}} }+\mathrm{x}_{\mathrm{n}} \right),\:\mathrm{n}\geqslant\mathrm{0} \\ $$$$\mathrm{Prove}:\:\mathrm{For}\:\mathrm{n}\geqslant\mathrm{1},\:\mathrm{x}_{\mathrm{n}} >\mathrm{x}_{\mathrm{n}+\mathrm{1}} >\sqrt{\mathrm{R}}\:\mathrm{and} \\ $$$$\mathrm{x}_{\mathrm{n}} −\sqrt{\mathrm{R}}\leqslant\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{n}} }\:\frac{\left(\mathrm{x}_{\mathrm{0}} −\sqrt{\mathrm{R}}\right)^{\mathrm{2}} }{\mathrm{x}_{\mathrm{0}} } \\ $$

Question Number 120036 Answers: 1 Comments: 0

Question Number 120035 Answers: 1 Comments: 0

Question Number 120029 Answers: 0 Comments: 2

Montrer que ∀x∈R cos(sinx)>sin(cosx)

$$\mathrm{Montrer}\:\mathrm{que}\:\forall\mathrm{x}\in\mathbb{R} \\ $$$$\mathrm{cos}\left(\mathrm{sinx}\right)>\mathrm{sin}\left(\mathrm{cosx}\right) \\ $$

Question Number 120028 Answers: 2 Comments: 0

Question Number 120025 Answers: 2 Comments: 0

calculate f^′ (x) 1) f(x) =∫_0 ^∞ ((cos(xt))/(t^2 +x^2 ))dt 2)f(x)=∫_0 ^∞ ((sin(xt^2 +(√2)))/(t^2 +x^2 +3))dt

$$\mathrm{calculate}\:\mathrm{f}^{'} \left(\mathrm{x}\right) \\ $$$$\left.\mathrm{1}\right)\:\mathrm{f}\left(\mathrm{x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{cos}\left(\mathrm{xt}\right)}{\mathrm{t}^{\mathrm{2}} +\mathrm{x}^{\mathrm{2}} }\mathrm{dt} \\ $$$$\left.\mathrm{2}\right)\mathrm{f}\left(\mathrm{x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{sin}\left(\mathrm{xt}^{\mathrm{2}} +\sqrt{\mathrm{2}}\right)}{\mathrm{t}^{\mathrm{2}} +\mathrm{x}^{\mathrm{2}} \:+\mathrm{3}}\mathrm{dt} \\ $$

Question Number 120016 Answers: 2 Comments: 0

Question Number 120008 Answers: 2 Comments: 2

Question Number 120006 Answers: 0 Comments: 0

Question Number 119997 Answers: 4 Comments: 0

solve for x,a∈R. (√(x^2 +ax+a^2 ))+(√(x^2 −ax+a^2 ))=1

$$\mathrm{solve}\:\mathrm{for}\:\mathrm{x},\mathrm{a}\in\boldsymbol{\mathrm{R}}. \\ $$$$\sqrt{\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{ax}}+\boldsymbol{{a}}^{\mathrm{2}} }+\sqrt{\boldsymbol{{x}}^{\mathrm{2}} −\boldsymbol{{ax}}+\boldsymbol{{a}}^{\mathrm{2}} }=\mathrm{1} \\ $$

Question Number 119996 Answers: 1 Comments: 0

{ ((x^3 +y^2 =a)),((x^2 +y^3 =b)) :} [solve for:x,y,a≠b∈R]

$$\begin{cases}{\boldsymbol{{x}}^{\mathrm{3}} +\boldsymbol{{y}}^{\mathrm{2}} =\boldsymbol{{a}}}\\{\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{y}}^{\mathrm{3}} =\boldsymbol{{b}}}\end{cases}\:\:\:\left[\boldsymbol{{solve}}\:\boldsymbol{{for}}:\mathrm{x},\mathrm{y},\mathrm{a}\neq\mathrm{b}\in\boldsymbol{\mathrm{R}}\right] \\ $$

Question Number 119989 Answers: 2 Comments: 0

Question Number 119979 Answers: 3 Comments: 1

Question Number 119977 Answers: 2 Comments: 0

lim_(x→∞) ((x^4 )^(1/5) (((x+1))^(1/5) −(x)^(1/5) ))=?

$$\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\sqrt[{\mathrm{5}}]{{x}^{\mathrm{4}} }\left(\sqrt[{\mathrm{5}}]{{x}+\mathrm{1}}−\sqrt[{\mathrm{5}}]{{x}}\:\right)\right)=? \\ $$

Question Number 119970 Answers: 3 Comments: 0

∫ (dx/(x^2 (√(25−x^2 )))) ?

$$\:\int\:\frac{{dx}}{{x}^{\mathrm{2}} \sqrt{\mathrm{25}−{x}^{\mathrm{2}} }}\:? \\ $$

Question Number 119969 Answers: 1 Comments: 0

Question Number 119965 Answers: 1 Comments: 0

Question Number 119961 Answers: 3 Comments: 0

Without L′Hopital rule lim_(x→π/3) ((sin (x−(π/3)))/(1−2cos x)) ?

$${Without}\:{L}'{Hopital}\:{rule}\: \\ $$$$\:\underset{{x}\rightarrow\pi/\mathrm{3}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:\left({x}−\frac{\pi}{\mathrm{3}}\right)}{\mathrm{1}−\mathrm{2cos}\:{x}}\:? \\ $$

Question Number 119960 Answers: 1 Comments: 0

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