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Question Number 33690    Answers: 1   Comments: 0

∫(dx/((x)^(1/3) +(√x)))

$$\int\frac{\boldsymbol{\mathrm{dx}}}{\sqrt[{\mathrm{3}}]{\boldsymbol{\mathrm{x}}}+\sqrt{\boldsymbol{\mathrm{x}}}} \\ $$

Question Number 33689    Answers: 2   Comments: 1

∫(x/(x^3 +1))dx

$$\int\frac{{x}}{{x}^{\mathrm{3}} +\mathrm{1}}{dx} \\ $$

Question Number 33688    Answers: 1   Comments: 0

given that f(x)=(1/2)(10^x +10^(−x) ) prove that 2f(x) f(y)=f(x+y)+f(x−y)

$$\boldsymbol{\mathrm{given}}\:\boldsymbol{\mathrm{that}}\: \\ $$$$\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}\right)=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{10}^{\boldsymbol{{x}}} +\mathrm{10}^{−\boldsymbol{{x}}} \right)\:\boldsymbol{\mathrm{prove}}\:\boldsymbol{\mathrm{that}} \\ $$$$\mathrm{2}\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}\right)\:\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{y}}\right)=\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}\right)+\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{y}}\right) \\ $$

Question Number 33681    Answers: 0   Comments: 0

Let a, b, c are positive real numbers such that (1/a) + (1/b) + (1/c) = 3 . Prove that : a + b + c + (4/(1 + (((abc)^2 ))^(1/3) )) ≥ 5

$${Let}\:\:{a},\:{b},\:{c}\:\:\:{are}\:\:{positive}\:{real}\:\:{numbers}\:\:{such}\:\:{that}\:\:\:\frac{\mathrm{1}}{{a}}\:+\:\frac{\mathrm{1}}{{b}}\:+\:\frac{\mathrm{1}}{{c}}\:\:=\:\:\mathrm{3}\:. \\ $$$${Prove}\:{that}\::\:\:\:{a}\:+\:{b}\:+\:{c}\:\:+\:\:\frac{\mathrm{4}}{\mathrm{1}\:+\:\sqrt[{\mathrm{3}}]{\left({abc}\right)^{\mathrm{2}} }}\:\:\:\geqslant\:\:\mathrm{5} \\ $$

Question Number 33677    Answers: 0   Comments: 1

calculate ∫_0 ^1 ((xlnx)/(x−1))dx .

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{xlnx}}{{x}−\mathrm{1}}{dx}\:. \\ $$

Question Number 33676    Answers: 0   Comments: 0

How fast is the height of a balloon changing when 500m away at an angle of π/4 rad and the angle is increasing by 0.2rad/min

$${How}\:{fast}\:{is}\:{the}\:{height}\:{of}\:{a}\:{balloon} \\ $$$${changing}\:{when}\:\mathrm{500}{m}\:{away}\:{at}\:{an} \\ $$$${angle}\:{of}\:\pi/\mathrm{4}\:{rad}\:{and}\:{the}\:{angle}\:{is} \\ $$$${increasing}\:{by}\:\mathrm{0}.\mathrm{2}{rad}/{min} \\ $$

Question Number 33671    Answers: 0   Comments: 0

Question Number 33663    Answers: 0   Comments: 0

Qn:(a) given A and B are disjoint sets shade in venn diagram (i)B∩C (ii)A−(B∪C) (iii)(B∪C)−A Qn:(b) Given that ∣A∣=19 ∣B∣=23 ∣C∣=24 ∣A∪B∣=30 ∣(A∩B)−C∣=5 ∣(B∩C)∣=10 ∣(A∪B)^′ ∣=20 and ∣C−(A∪B)∣=10 then find (i)∣(A∩C)∣ (ii)∣(B−′C)∣′ (iii)∣𝛍∣ (iv)∣(B′∪C)∩A′∣

$$\:\boldsymbol{\mathrm{Qn}}:\left(\boldsymbol{\mathrm{a}}\right) \\ $$$$\:\boldsymbol{\mathrm{given}}\:\boldsymbol{\mathrm{A}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{B}}\:\boldsymbol{\mathrm{are}}\:\boldsymbol{\mathrm{disjoint}}\:\boldsymbol{\mathrm{sets}} \\ $$$$\:\boldsymbol{\mathrm{shade}}\:\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{venn}}\:\boldsymbol{\mathrm{diagram}} \\ $$$$\:\left(\boldsymbol{\mathrm{i}}\right)\boldsymbol{\mathrm{B}}\cap\boldsymbol{\mathrm{C}}\:\left(\boldsymbol{\mathrm{ii}}\right)\boldsymbol{\mathrm{A}}−\left(\boldsymbol{\mathrm{B}}\cup\boldsymbol{\mathrm{C}}\right)\:\left(\boldsymbol{\mathrm{iii}}\right)\left(\boldsymbol{\mathrm{B}}\cup\boldsymbol{\mathrm{C}}\right)−\boldsymbol{\mathrm{A}} \\ $$$$\:\boldsymbol{\mathrm{Qn}}:\left(\boldsymbol{\mathrm{b}}\right) \\ $$$$\:\boldsymbol{\mathrm{G}}\mathrm{i}\boldsymbol{\mathrm{ven}}\:\boldsymbol{\mathrm{that}}\:\mid\boldsymbol{\mathrm{A}}\mid=\mathrm{19}\:\mid\boldsymbol{\mathrm{B}}\mid=\mathrm{23}\:\mid\boldsymbol{\mathrm{C}}\mid=\mathrm{24} \\ $$$$\:\mid\boldsymbol{\mathrm{A}}\cup\boldsymbol{\mathrm{B}}\mid=\mathrm{30}\:\mid\left(\boldsymbol{\mathrm{A}}\cap\boldsymbol{\mathrm{B}}\right)−\boldsymbol{\mathrm{C}}\mid=\mathrm{5}\:\mid\left(\boldsymbol{\mathrm{B}}\cap\boldsymbol{\mathrm{C}}\right)\mid=\mathrm{10} \\ $$$$\:\mid\left(\boldsymbol{\mathrm{A}}\cup\boldsymbol{\mathrm{B}}\right)^{'} \mid=\mathrm{20}\:\boldsymbol{\mathrm{and}}\:\mid\boldsymbol{\mathrm{C}}−\left(\boldsymbol{\mathrm{A}}\cup\boldsymbol{\mathrm{B}}\right)\mid=\mathrm{10} \\ $$$$\:\boldsymbol{\mathrm{then}}\:\boldsymbol{\mathrm{find}} \\ $$$$\:\left(\boldsymbol{\mathrm{i}}\right)\mid\left(\boldsymbol{\mathrm{A}}\cap\boldsymbol{\mathrm{C}}\right)\mid\:\left(\boldsymbol{\mathrm{ii}}\right)\mid\left(\boldsymbol{\mathrm{B}}−'\boldsymbol{\mathrm{C}}\right)\mid' \\ $$$$\:\left(\boldsymbol{\mathrm{iii}}\right)\mid\boldsymbol{\mu}\mid\:\left(\mathrm{iv}\right)\mid\left(\boldsymbol{\mathrm{B}}'\cup\boldsymbol{\mathrm{C}}\right)\cap\boldsymbol{\mathrm{A}}'\mid \\ $$

Question Number 33660    Answers: 4   Comments: 0

from sinhu=tan𝛝 prove that (i)tanh(u/2)=tan(𝛝/2) (ii)coshu=sec𝛝 (iii)u=log(sec𝛝+tan𝛝) (iv)tanhu=sin𝛝

$$\:\boldsymbol{\mathrm{from}}\:\boldsymbol{\mathrm{sinh}{u}}=\boldsymbol{\mathrm{tan}\vartheta} \\ $$$$\:\boldsymbol{\mathrm{prove}}\:\boldsymbol{\mathrm{that}} \\ $$$$\left(\boldsymbol{\mathrm{i}}\right)\boldsymbol{\mathrm{tanh}}\frac{\boldsymbol{\mathrm{u}}}{\mathrm{2}}=\boldsymbol{\mathrm{tan}}\frac{\boldsymbol{\vartheta}}{\mathrm{2}} \\ $$$$\:\left(\boldsymbol{\mathrm{ii}}\right)\boldsymbol{\mathrm{cosh}{u}}=\boldsymbol{\mathrm{sec}\vartheta} \\ $$$$\:\left(\boldsymbol{\mathrm{iii}}\right)\boldsymbol{\mathrm{u}}=\boldsymbol{\mathrm{log}}\left(\boldsymbol{\mathrm{sec}\vartheta}+\boldsymbol{\mathrm{tan}\vartheta}\right) \\ $$$$\:\left(\boldsymbol{\mathrm{iv}}\right)\boldsymbol{\mathrm{tanh}{u}}=\boldsymbol{\mathrm{sin}\vartheta} \\ $$

Question Number 33659    Answers: 1   Comments: 0

solve for x and y coshy−7sinhx=3 and coshy−3sinh^2 x=2

$$\:\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{x}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{y}} \\ $$$$\:\boldsymbol{\mathrm{cosh}{y}}−\mathrm{7}\boldsymbol{\mathrm{sinh}{x}}=\mathrm{3}\:\boldsymbol{\mathrm{and}} \\ $$$$\:\boldsymbol{\mathrm{cosh}{y}}−\mathrm{3}\boldsymbol{\mathrm{sinh}}^{\mathrm{2}} \boldsymbol{{x}}=\mathrm{2} \\ $$

Question Number 33658    Answers: 1   Comments: 1

If f(x)= x^3 −3x+1 then find number of different real solutions of f(f(x))=0 ?

$$\boldsymbol{{I}}{f}\:{f}\left({x}\right)=\:{x}^{\mathrm{3}} −\mathrm{3}{x}+\mathrm{1} \\ $$$${then}\:{find}\:{number}\:{of}\:{different}\:\:\:{real} \\ $$$${solutions}\:{of}\:{f}\left({f}\left({x}\right)\right)=\mathrm{0}\:? \\ $$

Question Number 33651    Answers: 1   Comments: 0

Let function f(x) be defined as f(x)= x^2 +bx+c , where b,c∈R . And f(1) − 2f(5) +f(9) =32. Find no. of ordered pairs (b,c) such that ∣f(x)∣≤8 ∀ x∈ [1,9] ?

$${Let}\:{function}\:{f}\left({x}\right)\:{be}\:{defined}\:{as}\: \\ $$$${f}\left({x}\right)=\:{x}^{\mathrm{2}} +{bx}+{c}\:,\:{where}\:{b},{c}\in{R}\:. \\ $$$${And}\:{f}\left(\mathrm{1}\right)\:−\:\mathrm{2}{f}\left(\mathrm{5}\right)\:+{f}\left(\mathrm{9}\right)\:=\mathrm{32}. \\ $$$${Find}\:{no}.\:{of}\:{ordered}\:{pairs}\:\left({b},{c}\right) \\ $$$${such}\:{that}\:\mid{f}\left({x}\right)\mid\leqslant\mathrm{8}\:\forall\:{x}\in\:\left[\mathrm{1},\mathrm{9}\right]\:? \\ $$

Question Number 33649    Answers: 1   Comments: 3

Consider f:R^+ →R such that f(3)=1 for a∈R^+ and f(x).f(y) + f((3/x)).f((3/y)) = 2f(xy) ∀ x,y ∈ R^+ . Then find f(x) ?

$${Consider}\:{f}:{R}^{+} \rightarrow{R}\:{such}\:{that} \\ $$$${f}\left(\mathrm{3}\right)=\mathrm{1}\:{for}\:{a}\in{R}^{+} \:{and}\: \\ $$$${f}\left({x}\right).{f}\left({y}\right)\:+\:{f}\left(\frac{\mathrm{3}}{{x}}\right).{f}\left(\frac{\mathrm{3}}{{y}}\right)\:=\:\mathrm{2}{f}\left({xy}\right) \\ $$$$\forall\:{x},{y}\:\in\:{R}^{+} .\:{Then}\:{find}\:{f}\left({x}\right)\:? \\ $$

Question Number 33644    Answers: 1   Comments: 0

∫(sec^2 x)e^(tanx) dx

$$\:\int\left(\boldsymbol{\mathrm{sec}}^{\mathrm{2}} \boldsymbol{{x}}\right)\boldsymbol{{e}}^{\boldsymbol{\mathrm{tan}{x}}} \boldsymbol{{dx}} \\ $$

Question Number 33643    Answers: 1   Comments: 0

∫((2x+3)/(x^2 +4))dx

$$\:\int\frac{\mathrm{2}\boldsymbol{{x}}+\mathrm{3}}{\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{4}}\boldsymbol{{dx}} \\ $$

Question Number 33629    Answers: 1   Comments: 2

Question Number 33628    Answers: 0   Comments: 4

Question Number 33622    Answers: 0   Comments: 0

Question Number 33619    Answers: 1   Comments: 3

∫x^(5/2) (1−x)^(3/2) dx

$$\int{x}^{\mathrm{5}/\mathrm{2}} \left(\mathrm{1}−{x}\right)^{\mathrm{3}/\mathrm{2}} {dx} \\ $$

Question Number 33616    Answers: 1   Comments: 4

please help me please is there any app for practicing calculus for a CBT exam.I mean one that has a timer so I can asses my speed.

$${please}\:{help}\:{me} \\ $$$$ \\ $$$${please}\:{is}\:{there}\:{any}\:{app}\:{for}\:{practicing} \\ $$$${calculus}\:{for}\:\:{a}\:{CBT}\:{exam}.{I}\:{mean} \\ $$$${one}\:{that}\:{has}\:{a}\:{timer}\:{so}\:{I}\:{can}\:{asses} \\ $$$${my}\:{speed}. \\ $$

Question Number 33599    Answers: 1   Comments: 2

calculatef(a)= ∫_(−a) ^a (dx/((t^2 +x^2 )^(3/2) )) with a>0 .

$${calculatef}\left({a}\right)=\:\:\int_{−{a}} ^{{a}} \:\:\:\:\frac{{dx}}{\left({t}^{\mathrm{2}} \:+{x}^{\mathrm{2}} \right)^{\frac{\mathrm{3}}{\mathrm{2}}} }\:\:{with}\:{a}>\mathrm{0}\:. \\ $$

Question Number 33597    Answers: 0   Comments: 0

study and give the graph of f(x) =e^(2/(lnx)) .

$${study}\:{and}\:{give}\:{the}\:{graph}\:{of}\:{f}\left({x}\right)\:={e}^{\frac{\mathrm{2}}{{lnx}}} \:. \\ $$

Question Number 33596    Answers: 0   Comments: 0

1) prove that ∀(a,b)∈R^2 ∣sinb −sina∣≤∣b−a∣ 2)let give the sequence x_0 =0 and x_(n+1) =a +(1/2)sin(x_n ) prove that for m≥n ∣x_m −x_n ∣ ≤ ((∣a∣)/2^(n−1) ) 3) prove that (x_n ) is convergent and its limit is solution of the equation x = a +(1/2) sinx .

$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\forall\left({a},{b}\right)\in{R}^{\mathrm{2}} \:\:\:\:\mid{sinb}\:−{sina}\mid\leqslant\mid{b}−{a}\mid \\ $$$$\left.\mathrm{2}\right){let}\:{give}\:{the}\:{sequence}\:\:{x}_{\mathrm{0}} =\mathrm{0}\:{and} \\ $$$${x}_{{n}+\mathrm{1}} ={a}\:+\frac{\mathrm{1}}{\mathrm{2}}{sin}\left({x}_{{n}} \right)\:{prove}\:{that}\:{for}\:{m}\geqslant{n} \\ $$$$\mid{x}_{{m}} \:−{x}_{{n}} \mid\:\leqslant\:\:\frac{\mid{a}\mid}{\mathrm{2}^{{n}−\mathrm{1}} } \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:\left({x}_{{n}} \right)\:{is}\:{convergent}\:{and}\:{its}\:{limit}\:{is}\:{solution} \\ $$$${of}\:{the}\:{equation}\:\:{x}\:=\:{a}\:+\frac{\mathrm{1}}{\mathrm{2}}\:{sinx}\:. \\ $$

Question Number 33595    Answers: 0   Comments: 0

find lim_(x→0) ((ln(1+sinx) −x(√(1−x)))/(sinx −shx)) .

$${find}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\frac{{ln}\left(\mathrm{1}+{sinx}\right)\:−{x}\sqrt{\mathrm{1}−{x}}}{{sinx}\:−{shx}}\:\:. \\ $$

Question Number 33594    Answers: 0   Comments: 1

calculate lim_(x→1^− ) (1/((1−x)^α ))(arcsinx −(π/2)) .

$${calculate}\:\:{lim}_{{x}\rightarrow\mathrm{1}^{−} } \:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\left(\mathrm{1}−{x}\right)^{\alpha} }\left({arcsinx}\:−\frac{\pi}{\mathrm{2}}\right)\:. \\ $$

Question Number 33593    Answers: 0   Comments: 0

calculate lim_(x→0) ((2(1−cosx)sinx −x^3 (1−x^2 )^(1/4) )/(sin^5 x −x^5 ))

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\:\frac{\mathrm{2}\left(\mathrm{1}−{cosx}\right){sinx}\:−{x}^{\mathrm{3}} \:\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\frac{\mathrm{1}}{\mathrm{4}}} }{{sin}^{\mathrm{5}} {x}\:−{x}^{\mathrm{5}} } \\ $$

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