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Question Number 38310    Answers: 1   Comments: 4

let f(x)=∫_0 ^(+∞) ((arctan(xt))/(1+t^2 ))dt with x≥0 1) calculate f^′ (x) then a simple form of f(x) 2) calculate ∫_0 ^(+∞) ((arctant)/(1+t^2 ))dt 3) calculate ∫_0 ^(+∞) ((arctan(2t))/(1+t^2 ))dt

$${let}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{+\infty} \:\:\:\frac{{arctan}\left({xt}\right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}\:\:{with}\:{x}\geqslant\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{'} \left({x}\right)\:{then}\:{a}\:{simple}\:{form}\:{of}\:\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{+\infty} \:\:\:\frac{{arctant}}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{+\infty} \:\:\frac{{arctan}\left(\mathrm{2}{t}\right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$

Question Number 38323    Answers: 1   Comments: 1

find Σ_(n=1) ^(+∞) ((4n)/((2n−1)^2 (2n+1)^2 ))

$${find}\:\sum_{{n}=\mathrm{1}} ^{+\infty} \:\:\:\frac{\mathrm{4}{n}}{\left(\mathrm{2}{n}−\mathrm{1}\right)^{\mathrm{2}} \left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 38296    Answers: 0   Comments: 7

i have a suggestion pls comment...we are all virtual friends common bond is mathematics so may know each other by posting our self photo...if administator give permission..

$${i}\:{have}\:{a}\:{suggestion}\:{pls}\:{comment}...{we}\:{are}\:{all} \\ $$$${virtual}\:{friends}\:{common}\:{bond}\:{is}\:{mathematics} \\ $$$${so}\:{may}\:{know}\:{each}\:{other}\:{by}\:{posting}\:{our}\:{self} \\ $$$${photo}...{if}\:{administator}\:{give}\:{permission}.. \\ $$

Question Number 38291    Answers: 1   Comments: 0

Question Number 38289    Answers: 1   Comments: 0

Question Number 38288    Answers: 1   Comments: 0

tan α−tan β = 2tan θ asin α−bsin β = lsin θ express sin α, sin β in terms of 𝛉.

$$\mathrm{tan}\:\alpha−\mathrm{tan}\:\beta\:=\:\mathrm{2tan}\:\theta \\ $$$${a}\mathrm{sin}\:\alpha−{b}\mathrm{sin}\:\beta\:=\:{l}\mathrm{sin}\:\theta \\ $$$${express}\:\mathrm{sin}\:\alpha,\:\mathrm{sin}\:\beta\:\:{in}\:{terms}\:{of}\:\boldsymbol{\theta}. \\ $$

Question Number 38286    Answers: 0   Comments: 1

(i) given the function f(t)=e^t and g(t)=lnt show that f○g(t)=g○f(t) (ii)if f(t)=at , g(t)=bt^2 +3 (fog)(2)=35 and (fog)(3)=75 find the value of a and b

$$\left(\boldsymbol{{i}}\right)\:\mathrm{given}\:\mathrm{the}\:\mathrm{function}\:\boldsymbol{{f}}\left(\boldsymbol{{t}}\right)=\boldsymbol{\mathrm{e}}^{\boldsymbol{{t}}} \:\:\boldsymbol{\mathrm{and}}\:\boldsymbol{{g}}\left(\boldsymbol{{t}}\right)=\boldsymbol{\mathrm{ln}{t}} \\ $$$$\boldsymbol{\mathrm{show}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{{f}}\circ\boldsymbol{{g}}\left(\boldsymbol{{t}}\right)=\boldsymbol{{g}}\circ\boldsymbol{{f}}\left(\boldsymbol{{t}}\right) \\ $$$$\left(\boldsymbol{{ii}}\right)\mathrm{if}\:\boldsymbol{{f}}\left(\boldsymbol{{t}}\right)=\boldsymbol{{at}}\:,\:\boldsymbol{{g}}\left(\boldsymbol{{t}}\right)=\boldsymbol{{bt}}^{\mathrm{2}} +\mathrm{3} \\ $$$$\left(\boldsymbol{{fog}}\right)\left(\mathrm{2}\right)=\mathrm{35}\:\boldsymbol{\mathrm{and}}\:\left(\boldsymbol{{fog}}\right)\left(\mathrm{3}\right)=\mathrm{75} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\boldsymbol{{a}}\:\mathrm{and}\:\boldsymbol{{b}} \\ $$

Question Number 38284    Answers: 1   Comments: 2

Find all the complex number in the rectangular form such that (z−1)^4 =−1

$${Find}\:{all}\:{the}\:{complex}\:{number}\:{in}\:{the} \\ $$$${rectangular}\:{form}\:{such}\:{that} \\ $$$$\left({z}−\mathrm{1}\right)^{\mathrm{4}} =−\mathrm{1} \\ $$$$ \\ $$

Question Number 38282    Answers: 1   Comments: 0

Question Number 38281    Answers: 1   Comments: 0

Question Number 38262    Answers: 1   Comments: 0

Question Number 38252    Answers: 0   Comments: 2

if x^2 + 3xy − y^2 = 3 find (dy/dx) at point (1,1) hence differentiate ((sin x)/(1 + x)) with respect to x.

$${if}\:{x}^{\mathrm{2}} \:+\:\mathrm{3}{xy}\:−\:{y}^{\mathrm{2}} \:=\:\mathrm{3}\:{find}\: \\ $$$$\frac{{dy}}{{dx}}\:{at}\:{point}\:\left(\mathrm{1},\mathrm{1}\right)\:{hence} \\ $$$${differentiate}\:\frac{{sin}\:{x}}{\mathrm{1}\:+\:{x}}\:{with}\:{respect} \\ $$$${to}\:{x}. \\ $$

Question Number 38250    Answers: 0   Comments: 0

It is given that the first term of a GP is the last term of an AP. the second term of the AP is the third term of the GP..detemine the Geometric mean of the GP is the fourth term of the GP is 16.

$$\:\:{It}\:{is}\:{given}\:{that}\:{the}\:{first}\:{term}\:{of} \\ $$$${a}\:{GP}\:\:{is}\:{the}\:{last}\:{term}\:{of}\:{an}\:{AP}. \\ $$$${the}\:{second}\:{term}\:{of}\:{the}\:{AP}\:{is}\:{the} \\ $$$${third}\:{term}\:{of}\:{the}\:{GP}..{detemine} \\ $$$${the}\:{Geometric}\:{mean}\:{of}\:{the}\:{GP}\:{is}\: \\ $$$$\:{the}\:{fourth}\:{term}\:{of}\:{the}\:{GP}\:{is}\:\mathrm{16}. \\ $$

Question Number 38247    Answers: 2   Comments: 2

Question Number 38235    Answers: 0   Comments: 4

A man 2m 50cm tall stands a distance of 3m in front of a large vertical plane mirror. i)what is the shortest length of the mirror that will enable the man see himself fully? ii)what is the answer of the above if the man were 5m away?

$${A}\:{man}\:\mathrm{2}{m}\:\mathrm{50}{cm}\:{tall}\:{stands}\:{a} \\ $$$${distance}\:{of}\:\mathrm{3}{m}\:{in}\:{front}\:{of}\:{a}\:{large} \\ $$$${vertical}\:{plane}\:{mirror}. \\ $$$$\left.{i}\right){what}\:{is}\:{the}\:{shortest}\:{length}\:{of}\:{the} \\ $$$${mirror}\:{that}\:{will}\:{enable}\:{the}\:{man}\:{see} \\ $$$${himself}\:{fully}? \\ $$$$\left.{ii}\right){what}\:{is}\:{the}\:{answer}\:{of}\:{the}\:{above} \\ $$$${if}\:{the}\:{man}\:{were}\:\mathrm{5}{m}\:{away}? \\ $$

Question Number 38261    Answers: 1   Comments: 0

Question Number 38232    Answers: 4   Comments: 1

Differentiate tan^(−1) ((((√(1+x^2 ))−1)/x)) without using any trigonometric substitution !

$$\mathrm{Differentiate}\: \\ $$$$\mathrm{tan}^{−\mathrm{1}} \left(\frac{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }−\mathrm{1}}{{x}}\right)\:\: \\ $$$${without}\:{using}\:{any}\:{trigonometric}\: \\ $$$${substitution}\:! \\ $$

Question Number 38222    Answers: 0   Comments: 1

If U={−5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5} A={x/x^2 =25, x ∈ Z} B={x/x^2 +5=9, x ∈ Z} and C={x/−2≤ x ≤ 2, x ∈ Z} then (A ∩ B ∩ C)^c ∩ (A△B)^c =?

$$\mathrm{If}\:\mathbb{U}=\left\{−\mathrm{5},\:−\mathrm{4},\:−\mathrm{3},\:−\mathrm{2},\:−\mathrm{1},\:\mathrm{0},\:\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:\mathrm{4},\:\mathrm{5}\right\} \\ $$$$\mathrm{A}=\left\{{x}/{x}^{\mathrm{2}} =\mathrm{25},\:{x}\:\in\:\mathrm{Z}\right\} \\ $$$$\mathrm{B}=\left\{{x}/{x}^{\mathrm{2}} +\mathrm{5}=\mathrm{9},\:{x}\:\in\:\mathrm{Z}\right\}\:\mathrm{and} \\ $$$$\mathrm{C}=\left\{{x}/−\mathrm{2}\leqslant\:{x}\:\leqslant\:\mathrm{2},\:{x}\:\in\:\mathrm{Z}\right\}\:\mathrm{then} \\ $$$$\left(\mathrm{A}\:\cap\:\mathrm{B}\:\cap\:\mathrm{C}\right)^{\mathrm{c}} \:\cap\:\left(\mathrm{A}\bigtriangleup\mathrm{B}\right)^{\mathrm{c}} =? \\ $$

Question Number 38211    Answers: 0   Comments: 2

let x>0 and F(x)= ∫_0 ^(+∞) ((arctan(xt^2 ))/(1+t^2 ))dt 1) find a simple form of F(x) 2)find the value of ∫_0 ^∞ ((arctan(2t^2 ))/(1+t^2 ))dt 3)find the value of ∫_0 ^∞ ((arctan(3t^2 ))/(1+t^2 ))dt.

$${let}\:{x}>\mathrm{0}\:{and}\:{F}\left({x}\right)=\:\int_{\mathrm{0}} ^{+\infty} \:\frac{{arctan}\left({xt}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{simple}\:{form}\:{of}\:{F}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{arctan}\left(\mathrm{2}{t}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \frac{{arctan}\left(\mathrm{3}{t}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}. \\ $$

Question Number 38210    Answers: 2   Comments: 4

let f(a)= ∫_0 ^π (dθ/(a +sin^2 θ)) (a from R) 1) find f(a) 2)calculate g(a)= ∫_0 ^π (dθ/((a+sin^2 θ)^2 )) 3)calculate ∫_0 ^π (dθ/(1+sin^2 θ)) and ∫_0 ^π (dθ/(2+sin^2 θ)) 4) calculate ∫_0 ^π (dθ/((3 +sin^2 θ)^2 )) .

$${let}\:{f}\left({a}\right)=\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{d}\theta}{{a}\:+{sin}^{\mathrm{2}} \theta}\:\:\:\left({a}\:{from}\:{R}\right) \\ $$$$\left.\mathrm{1}\right)\:{find}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right){calculate}\:{g}\left({a}\right)=\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{d}\theta}{\left({a}+{sin}^{\mathrm{2}} \theta\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right){calculate}\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{d}\theta}{\mathrm{1}+{sin}^{\mathrm{2}} \theta}\:{and}\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{{d}\theta}{\mathrm{2}+{sin}^{\mathrm{2}} \theta} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{d}\theta}{\left(\mathrm{3}\:+{sin}^{\mathrm{2}} \theta\right)^{\mathrm{2}} }\:. \\ $$

Question Number 38209    Answers: 0   Comments: 2

let f(x)=e^(−x) cosx developp f at fourier serie 1) find the value of Σ_(n=−∞) ^(+∞) (((−1)^n )/(1+n^2 )) 2) calculate Σ_(n=0) ^∞ (1/(n^2 +1)) .

$${let}\:{f}\left({x}\right)={e}^{−{x}} {cosx} \\ $$$${developp}\:{f}\:{at}\:{fourier}\:{serie} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{value}\:{of}\:\sum_{{n}=−\infty} ^{+\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} }{\mathrm{1}+{n}^{\mathrm{2}} } \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{1}}{{n}^{\mathrm{2}} \:+\mathrm{1}}\:. \\ $$

Question Number 38208    Answers: 0   Comments: 3

let f(x)=ch(αx) developp f at fourier serie. (f 2π periodic even)

$${let}\:{f}\left({x}\right)={ch}\left(\alpha{x}\right)\: \\ $$$${developp}\:{f}\:{at}\:{fourier}\:{serie}. \\ $$$$\left({f}\:\mathrm{2}\pi\:{periodic}\:{even}\right) \\ $$

Question Number 38207    Answers: 0   Comments: 1

prove that coth(x)−(1/x) =Σ_(n=1) ^∞ ((2x)/(x^2 +n^2 π^2 )) (x≠0)

$${prove}\:{that}\:{coth}\left({x}\right)−\frac{\mathrm{1}}{{x}}\:=\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\mathrm{2}{x}}{{x}^{\mathrm{2}} \:+{n}^{\mathrm{2}} \pi^{\mathrm{2}} } \\ $$$$\left({x}\neq\mathrm{0}\right) \\ $$

Question Number 38206    Answers: 1   Comments: 1

calculate lim_(x→0) ((x coth(x)−1)/x^2 )

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\frac{{x}\:{coth}\left({x}\right)−\mathrm{1}}{{x}^{\mathrm{2}} } \\ $$

Question Number 38205    Answers: 0   Comments: 0

if (1/(sinx)) =Σ_(n=1) ^∞ a_n sin(nx) find the values of a_n .

$${if}\:\:\frac{\mathrm{1}}{{sinx}}\:=\sum_{{n}=\mathrm{1}} ^{\infty} \:{a}_{{n}} {sin}\left({nx}\right)\:\:{find}\:{the}\:{values}\:{of} \\ $$$${a}_{{n}} . \\ $$

Question Number 38204    Answers: 0   Comments: 1

if (1/(cosx)) =(a_0 /2) +Σ_(n=1) ^∞ a_n cos(nx) calculate a_0 and a_n

$${if}\:\:\frac{\mathrm{1}}{{cosx}}\:=\frac{{a}_{\mathrm{0}} }{\mathrm{2}}\:+\sum_{{n}=\mathrm{1}} ^{\infty} \:{a}_{{n}} {cos}\left({nx}\right) \\ $$$${calculate}\:{a}_{\mathrm{0}} \:{and}\:{a}_{{n}} \\ $$

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