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Question Number 61979    Answers: 0   Comments: 1

find ∫_0 ^1 ln(x)ln(1+x) dx

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left({x}\right){ln}\left(\mathrm{1}+{x}\right)\:{dx} \\ $$

Question Number 61978    Answers: 0   Comments: 5

let f(x) =∫_0 ^1 ln(1−xt^3 )dt with ∣x∣<1 1) find a explicit form of f(x) 2)calculate ∫_0 ^1 ln(1−(1/(√2))t^3 )dt 3) calculate A(θ) =∫_0 ^1 ln(1−sinθ t^3 )dt with 0<θ<(π/2)

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}−{xt}^{\mathrm{3}} \right){dt}\:\:{with}\:\:\mid{x}\mid<\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}−\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}{t}^{\mathrm{3}} \right){dt} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:{A}\left(\theta\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}−{sin}\theta\:{t}^{\mathrm{3}} \right){dt}\:\:{with}\:\:\mathrm{0}<\theta<\frac{\pi}{\mathrm{2}} \\ $$

Question Number 61976    Answers: 0   Comments: 2

let A_n = ∫_(1/n) ^n ((arctan(x^2 +y^2 ))/(x^2 +y^2 )) dxdy 1) calculate A_n 2) find lim_(n→∞) A_n

$${let}\:{A}_{{n}} =\:\int_{\frac{\mathrm{1}}{{n}}} ^{{n}} \:\:\:\frac{{arctan}\left({x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }\:{dxdy}\:\: \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow\infty} \:{A}_{{n}} \\ $$

Question Number 61966    Answers: 2   Comments: 4

∫(1/(e^(2x) −e^(−2x) )) dx

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

Question Number 61954    Answers: 0   Comments: 1

Question Number 61952    Answers: 1   Comments: 1

Question Number 61948    Answers: 0   Comments: 0

The vectors a, b, c are equal in length and taken pairwise, they make equal angles. If a=i+j, b=j+k and c makes an obtuse angle with X−axis, then c=

$$\mathrm{The}\:\mathrm{vectors}\:\boldsymbol{\mathrm{a}},\:\boldsymbol{\mathrm{b}},\:\boldsymbol{\mathrm{c}}\:\mathrm{are}\:\mathrm{equal}\:\mathrm{in}\:\mathrm{length} \\ $$$$\mathrm{and}\:\mathrm{taken}\:\mathrm{pairwise},\:\mathrm{they}\:\mathrm{make}\:\mathrm{equal} \\ $$$$\mathrm{angles}.\:\mathrm{If}\:\:\boldsymbol{\mathrm{a}}=\boldsymbol{\mathrm{i}}+\boldsymbol{\mathrm{j}},\:\:\boldsymbol{\mathrm{b}}=\boldsymbol{\mathrm{j}}+\boldsymbol{\mathrm{k}}\:\:\mathrm{and}\:\:\boldsymbol{\mathrm{c}}\:\mathrm{makes} \\ $$$$\mathrm{an}\:\mathrm{obtuse}\:\mathrm{angle}\:\mathrm{with}\:{X}−\mathrm{axis},\:\mathrm{then}\:\boldsymbol{\mathrm{c}}= \\ $$

Question Number 61938    Answers: 0   Comments: 0

Question Number 61937    Answers: 1   Comments: 5

find the value of Σ_(n = 0) ^∞ ((n^3 + 5)/(n!))

$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:\:\:\:\underset{\mathrm{n}\:=\:\mathrm{0}} {\overset{\infty} {\sum}}\:\:\frac{\mathrm{n}^{\mathrm{3}} \:+\:\mathrm{5}}{\mathrm{n}!} \\ $$

Question Number 61934    Answers: 1   Comments: 2

Answer: 0^0 =?

$$\mathrm{Answer}:\:\mathrm{0}^{\mathrm{0}} =? \\ $$

Question Number 61923    Answers: 1   Comments: 0

Find all solutions of x^3 − 12x + 8 = 0

$${Find}\:\:{all}\:\:{solutions}\:\:{of} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:{x}^{\mathrm{3}} \:−\:\mathrm{12}{x}\:+\:\mathrm{8}\:=\:\:\mathrm{0} \\ $$

Question Number 61922    Answers: 1   Comments: 3

Find the value of: Σ_(n = 1) ^∞ ((n^2 + 1)/(n + 2)). (x^n /(n!))

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}:\:\:\:\:\:\underset{\mathrm{n}\:=\:\mathrm{1}} {\overset{\infty} {\sum}}\:\:\frac{\mathrm{n}^{\mathrm{2}} \:+\:\mathrm{1}}{\mathrm{n}\:+\:\mathrm{2}}.\:\frac{\mathrm{x}^{\mathrm{n}} }{\mathrm{n}!} \\ $$

Question Number 61921    Answers: 1   Comments: 1

let A =∫_(−∞) ^(+∞) ((x+1)/((x^2 +x+1)( x^2 −2i)))dx 1) calculate A 2) extract Re(A) and Im(A) and determine its values (i^2 =−1)

$${let}\:{A}\:=\int_{−\infty} ^{+\infty} \:\:\:\:\:\frac{{x}+\mathrm{1}}{\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)\left(\:{x}^{\mathrm{2}} \:−\mathrm{2}{i}\right)}{dx} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A} \\ $$$$\left.\mathrm{2}\right)\:{extract}\:{Re}\left({A}\right)\:{and}\:{Im}\left({A}\right)\:{and}\:{determine}\:{its}\:{values}\:\:\:\left({i}^{\mathrm{2}} =−\mathrm{1}\right) \\ $$

Question Number 61915    Answers: 1   Comments: 0

1+iw+(iw)^2 +(iw)^3 +.........(iw)^(989) =? ans= (2/(1−iw)) answer is correct. pls help .. how to do this? TIA

$$\mathrm{1}+{iw}+\left({iw}\right)^{\mathrm{2}} +\left({iw}\right)^{\mathrm{3}} +.........\left({iw}\right)^{\mathrm{989}} =? \\ $$$$ \\ $$$${ans}=\:\:\:\:\frac{\mathrm{2}}{\mathrm{1}−{iw}}\:\:\:\:\:\:{answer}\:{is}\:{correct}. \\ $$$${pls}\:{help}\:..\:{how}\:{to}\:{do}\:{this}? \\ $$$${TIA} \\ $$

Question Number 61912    Answers: 0   Comments: 1

Question Number 61907    Answers: 1   Comments: 1

Question Number 61986    Answers: 1   Comments: 0

Find the area bounded by y(x+2)=x^4 , x=0,y=0 and x=3

$${Find}\:{the}\:{area}\:{bounded}\:{by}\:{y}\left({x}+\mathrm{2}\right)={x}^{\mathrm{4}} , \\ $$$${x}=\mathrm{0},{y}=\mathrm{0}\:{and}\:{x}=\mathrm{3} \\ $$

Question Number 61902    Answers: 0   Comments: 3

Question Number 61895    Answers: 0   Comments: 3

let A = (((1 −1)),((0 1)) ) 1) calculate A^n 2) find e^A ,e^(−A) 3) determine e^(iA) then cosA and sinA .

$${let}\:{A}\:=\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:−\mathrm{1}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}^{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\:{e}^{{A}} \:\:,{e}^{−{A}} \\ $$$$\left.\mathrm{3}\right)\:{determine}\:{e}^{{iA}} \:\:\:{then}\:\:{cosA}\:\:{and}\:{sinA}\:. \\ $$

Question Number 61892    Answers: 0   Comments: 1

(1/4) of (2/5)

$$\frac{\mathrm{1}}{\mathrm{4}}\:\mathrm{of}\:\frac{\mathrm{2}}{\mathrm{5}} \\ $$

Question Number 61886    Answers: 1   Comments: 0

a+bi=((2+i)/(1−i)) find 2(a^2 +b^2 )

$${a}+{bi}=\frac{\mathrm{2}+{i}}{\mathrm{1}−{i}} \\ $$$${find}\: \\ $$$$\mathrm{2}\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} \right) \\ $$

Question Number 61885    Answers: 0   Comments: 0

let I =∫_(−∞) ^(+∞) (dx/((x+i)^n )) and J =∫_(−∞) ^(+∞) (dx/((x−i)^n )) 1) calculate I and J interms of n 2) find thevalue of integral A_n =∫_(−∞) ^(+∞) (( cos(narctan((1/x))))/((1+x^2 )^(n/2) ))dx

$${let}\:{I}\:=\int_{−\infty} ^{+\infty} \:\:\frac{{dx}}{\left({x}+{i}\right)^{{n}} }\:\:{and}\:{J}\:=\int_{−\infty} ^{+\infty} \:\:\frac{{dx}}{\left({x}−{i}\right)^{{n}} } \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{I}\:{and}\:{J}\:{interms}\:{of}\:{n} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{thevalue}\:{of}\:{integral} \\ $$$${A}_{{n}} \:\:=\int_{−\infty} ^{+\infty} \:\:\:\frac{\:{cos}\left({narctan}\left(\frac{\mathrm{1}}{{x}}\right)\right)}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\frac{{n}}{\mathrm{2}}} }{dx}\:\:\:\: \\ $$$$ \\ $$

Question Number 61884    Answers: 0   Comments: 3

let f_n (a) =∫_(−∞) ^(+∞) ((cos(nx))/((x^2 +x +a)^2 ))dx with a≥1 1) find a explicit form of f_n (a) 2)study the convervenge of Σ f_n (a) 3) determine also g_n (a) = ∫_(−∞) ^(+∞) ((cos(nx))/((x^2 +x+a)^3 ))dx study the convergence of Σ gn(a)

$${let}\:{f}_{{n}} \left({a}\right)\:=\int_{−\infty} ^{+\infty} \:\:\:\frac{{cos}\left({nx}\right)}{\left({x}^{\mathrm{2}} +{x}\:\:+{a}\right)^{\mathrm{2}} }{dx}\:\:\:\:{with}\:\:\:{a}\geqslant\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}_{{n}} \left({a}\right) \\ $$$$\left.\mathrm{2}\right){study}\:{the}\:{convervenge}\:{of}\:\Sigma\:{f}_{{n}} \left({a}\right) \\ $$$$\left.\mathrm{3}\right)\:{determine}\:{also}\:{g}_{{n}} \left({a}\right)\:=\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{cos}\left({nx}\right)}{\left({x}^{\mathrm{2}} \:+{x}+{a}\right)^{\mathrm{3}} }{dx} \\ $$$${study}\:{the}\:{convergence}\:{of}\:\Sigma\:{gn}\left({a}\right) \\ $$

Question Number 61874    Answers: 0   Comments: 1

∫((xln(x)−x)/(ln^3 (x))) dx

$$\int\frac{{xln}\left({x}\right)−{x}}{{ln}^{\mathrm{3}} \left({x}\right)}\:{dx} \\ $$

Question Number 61873    Answers: 1   Comments: 2

((30x^8 y^(12) ))^(1/3) /^4 (√(6x^2 y^9 z)) simplifh this question

$$\sqrt[{\mathrm{3}}]{\mathrm{30x}^{\mathrm{8}} \mathrm{y}^{\mathrm{12}} }/^{\mathrm{4}} \sqrt{\mathrm{6x}^{\mathrm{2}} \mathrm{y}^{\mathrm{9}} \mathrm{z}}\:\:\:\:\mathrm{simplifh}\:\mathrm{this}\:\mathrm{question} \\ $$

Question Number 61867    Answers: 0   Comments: 0

a player kicked a football at angel 30 with the ground towards an empty goal post of hegith 3.4m the ball hits the crossbar of the goal post 30m away from where the ball was kicked. Take g=9.8m/s. Find the intial velocity u of the ball?.What is time taken for the ball to hit the crossbar?

$$\:{a}\:{player}\:{kicked}\:{a}\:{football}\:{at}\:{angel}\:\mathrm{30}\:{with}\:{the}\:{ground}\:{towards}\:{an}\:{empty}\:{goal}\:{post}\:{of}\:{hegith}\:\mathrm{3}.\mathrm{4}{m}\:{the}\:{ball}\:{hits}\:{the}\:{crossbar}\:{of}\:{the}\:{goal}\:{post}\:\mathrm{30}{m}\:{away}\:{from}\:{where}\:{the}\:{ball}\:{was}\:{kicked}.\:{Take}\:{g}=\mathrm{9}.\mathrm{8}{m}/{s}.\:{Find}\:{the}\:{intial}\:{velocity}\:{u}\:{of}\:{the}\:{ball}?.{What}\:{is}\:{time}\:{taken}\:{for}\:{the}\:{ball}\:{to}\:{hit}\:{the}\:{crossbar}? \\ $$

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