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Question Number 28824    Answers: 0   Comments: 0

by using residus theorem find the value of A_n = ∫_0 ^∞ (dx/(1+x^n )) with n integr and n≥2.

$${by}\:{using}\:{residus}\:{theorem}\:{find}\:{the}\:{value}\:{of} \\ $$$${A}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\mathrm{1}+{x}^{{n}} }\:\:{with}\:{n}\:{integr}\:{and}\:{n}\geqslant\mathrm{2}. \\ $$

Question Number 28823    Answers: 0   Comments: 0

find I = ∫_(−∞) ^(+∞) (((x−1)cosx)/(x^2 −2x+2))dx and J= ∫_(−∞) ^(+∞) (((x−1)sinx)/(x^2 −2x +2)) dx.

$${find}\:\:{I}\:=\:\int_{−\infty} ^{+\infty} \:\:\frac{\left({x}−\mathrm{1}\right){cosx}}{{x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{2}}{dx}\:{and} \\ $$$${J}=\:\int_{−\infty} ^{+\infty} \:\:\frac{\left({x}−\mathrm{1}\right){sinx}}{{x}^{\mathrm{2}} −\mathrm{2}{x}\:+\mathrm{2}}\:{dx}. \\ $$

Question Number 28821    Answers: 0   Comments: 0

find the value of ∫_0 ^(2π) ((4 cos(4θ))/(5−4cosθ)) dθ .

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\frac{\mathrm{4}\:{cos}\left(\mathrm{4}\theta\right)}{\mathrm{5}−\mathrm{4}{cos}\theta}\:{d}\theta\:. \\ $$

Question Number 28820    Answers: 0   Comments: 0

find the value of ∫_0 ^1 ((ln(1−(t^2 /4)))/t^2 )dt.

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{ln}\left(\mathrm{1}−\frac{{t}^{\mathrm{2}} }{\mathrm{4}}\right)}{{t}^{\mathrm{2}} }{dt}. \\ $$

Question Number 28819    Answers: 0   Comments: 0

let give f(x)= ∫_0 ^1 ((ln(1−x^2 t^2 ))/t^2 )dt with ∣x∣<1 by using derivation under ∫ find the value of f(x).

$${let}\:{give}\:{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}\left(\mathrm{1}−{x}^{\mathrm{2}} {t}^{\mathrm{2}} \right)}{{t}^{\mathrm{2}} }{dt}\:\:\:\:{with}\:\mid{x}\mid<\mathrm{1} \\ $$$${by}\:{using}\:{derivation}\:{under}\:\int\:\:{find}\:{the}\:{value}\:{of}\:{f}\left({x}\right). \\ $$

Question Number 28818    Answers: 0   Comments: 0

prove that (π/(4cos(((πα)/2))))=Σ_(p=0) ^∞ (((2p+1)(−1)^p )/((2p+1)^2 −α^2 )) α ∈R−Z.

$${prove}\:{that}\:\:\frac{\pi}{\mathrm{4}{cos}\left(\frac{\pi\alpha}{\mathrm{2}}\right)}=\sum_{{p}=\mathrm{0}} ^{\infty} \:\:\:\:\frac{\left(\mathrm{2}{p}+\mathrm{1}\right)\left(−\mathrm{1}\right)^{{p}} }{\left(\mathrm{2}{p}+\mathrm{1}\right)^{\mathrm{2}} \:−\alpha^{\mathrm{2}} } \\ $$$$\alpha\:\in{R}−{Z}. \\ $$

Question Number 28817    Answers: 0   Comments: 0

let give f(x)=e^(iαx) 2π prriodic and α ∈R−Z developp f at fourier series.

$${let}\:{give}\:{f}\left({x}\right)={e}^{{i}\alpha{x}} \:\:\mathrm{2}\pi\:{prriodic}\:{and}\:\alpha\:\in{R}−{Z} \\ $$$${developp}\:{f}\:{at}\:{fourier}\:{series}. \\ $$

Question Number 28816    Answers: 0   Comments: 0

find the value of I= ∫_0 ^π (dθ/(1+cos^4 θ)) .

$${find}\:{the}\:{value}\:{of}\:\:{I}=\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{d}\theta}{\mathrm{1}+{cos}^{\mathrm{4}} \theta}\:. \\ $$

Question Number 28815    Answers: 1   Comments: 0

find the value of ∫_0 ^π (dx/(2cos^2 x +sin^2 x)) .

$${find}\:{the}\:{value}\:{of}\:\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{dx}}{\mathrm{2}{cos}^{\mathrm{2}} {x}\:+{sin}^{\mathrm{2}} {x}}\:. \\ $$

Question Number 28814    Answers: 0   Comments: 2

find Σ_(n=1) ^∞ (1/(1^3 +2^3 +...+n^3 )) .

$${find}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\:\frac{\mathrm{1}}{\mathrm{1}^{\mathrm{3}} \:+\mathrm{2}^{\mathrm{3}} +...+{n}^{\mathrm{3}} }\:. \\ $$

Question Number 28813    Answers: 0   Comments: 0

let give F(t)=∫_0 ^∞ ((sin(x^2 ))/x^2 ) e^(−tx^2 ) dx with t>0 find (dF/dt)(t).

$${let}\:{give}\:{F}\left({t}\right)=\int_{\mathrm{0}} ^{\infty} \frac{{sin}\left({x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} }\:{e}^{−{tx}^{\mathrm{2}} } {dx}\:\:{with}\:{t}>\mathrm{0} \\ $$$${find}\:\:\frac{{dF}}{{dt}}\left({t}\right). \\ $$$$ \\ $$

Question Number 28812    Answers: 0   Comments: 0

find the value of ∫_0 ^∞ (((1−e^(−x) )sinx)/x^2 )dx

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\frac{\left(\mathrm{1}−{e}^{−{x}} \right){sinx}}{{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 28811    Answers: 0   Comments: 1

find ∫_0 ^∞ ln(1+e^(−xt) )dx with t>0 then give the value of ∫_0 ^∞ ln(1+e^(−x) )dx.

$${find}\:\int_{\mathrm{0}} ^{\infty} {ln}\left(\mathrm{1}+{e}^{−{xt}} \right){dx}\:{with}\:{t}>\mathrm{0}\:{then}\:{give}\:{the}\:{value}\:{of} \\ $$$$\int_{\mathrm{0}} ^{\infty} {ln}\left(\mathrm{1}+{e}^{−{x}} \right){dx}. \\ $$

Question Number 28808    Answers: 2   Comments: 0

Question Number 28806    Answers: 1   Comments: 0

If n(A)=15 and n(B)=25, (a) What are the greatest and least values of n(AuB)? (b) What are the greatest and least value of n(AnB)? (c) Draw Venn diagrams to illustrate the four situations in (a) and (b) above

$$\mathrm{If}\:\mathrm{n}\left(\mathrm{A}\right)=\mathrm{15}\:\mathrm{and}\:\mathrm{n}\left(\mathrm{B}\right)=\mathrm{25},\: \\ $$$$\left(\mathrm{a}\right)\:\mathrm{What}\:\mathrm{are}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{and}\:\mathrm{least}\:\mathrm{values}\:\mathrm{of}\:\mathrm{n}\left(\mathrm{AuB}\right)? \\ $$$$\left(\mathrm{b}\right)\:\mathrm{What}\:\mathrm{are}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{and}\:\mathrm{least}\:\mathrm{value}\:\mathrm{of}\:\mathrm{n}\left(\mathrm{AnB}\right)? \\ $$$$\left(\mathrm{c}\right)\:\mathrm{Draw}\:\mathrm{Venn}\:\mathrm{diagrams}\:\mathrm{to}\:\mathrm{illustrate}\:\mathrm{the}\:\mathrm{four}\: \\ $$$$\:\:\:\:\:\:\:\mathrm{situations}\:\mathrm{in}\:\left(\mathrm{a}\right)\:\mathrm{and}\:\left(\mathrm{b}\right)\:\mathrm{above} \\ $$

Question Number 28805    Answers: 1   Comments: 0

In a competition, a school awarded medals in different categories. 36 medals in dance,12 in dramatics and 18 medals in music.If these medals went to total 45,and only 4 persons got medals in all three catogories.Using set notations, how many received in exactly two of these categories?

$${In}\:{a}\:{competition},\:{a}\:{school}\:{awarded} \\ $$$${medals}\:{in}\:{different}\:{categories}. \\ $$$$\mathrm{36}\:{medals}\:{in}\:{dance},\mathrm{12}\:{in}\:{dramatics} \\ $$$${and}\:\mathrm{18}\:{medals}\:{in}\:{music}.{If}\:{these} \\ $$$${medals}\:{went}\:{to}\:{total}\:\mathrm{45},{and}\:{only} \\ $$$$\mathrm{4}\:{persons}\:{got}\:{medals}\:{in}\:{all}\:{three} \\ $$$${catogories}.{Using}\:{set}\:{notations}, \\ $$$${how}\:{many}\:{received}\:{in}\:{exactly} \\ $$$${two}\:{of}\:{these}\:{categories}? \\ $$

Question Number 28779    Answers: 3   Comments: 1

Question Number 28770    Answers: 1   Comments: 6

lim_(x→+∞) ((5x^4 −10x^2 +1)/(−3x^3 +10x^2 +50))

$$\underset{{x}\rightarrow+\infty} {{lim}}\:\frac{\mathrm{5}{x}^{\mathrm{4}} −\mathrm{10}{x}^{\mathrm{2}} +\mathrm{1}}{−\mathrm{3}{x}^{\mathrm{3}} +\mathrm{10}{x}^{\mathrm{2}} +\mathrm{50}} \\ $$

Question Number 28762    Answers: 1   Comments: 6

Question Number 28739    Answers: 1   Comments: 0

if S_n =((a(r^n −1))/(r−1)) make r the subject of formula

$${if}\:{S}_{{n}} =\frac{{a}\left({r}^{{n}} −\mathrm{1}\right)}{{r}−\mathrm{1}}\: \\ $$$$ \\ $$$${make}\:{r}\:{the}\:{subject}\:{of}\:{formula} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 28709    Answers: 1   Comments: 0

Question Number 28708    Answers: 0   Comments: 0

If θ = log_e {tan(((3π)/8))} , prove that 3 tanh(2θ) = 2(√2)

$$\mathrm{If}\:\:\:\theta\:\:=\:\:\mathrm{log}_{\mathrm{e}} \left\{\mathrm{tan}\left(\frac{\mathrm{3}\pi}{\mathrm{8}}\right)\right\}\:,\:\:\:\mathrm{prove}\:\mathrm{that}\:\:\:\:\:\:\:\:\:\:\:\mathrm{3}\:\mathrm{tanh}\left(\mathrm{2}\theta\right)\:=\:\mathrm{2}\sqrt{\mathrm{2}} \\ $$

Question Number 28706    Answers: 1   Comments: 7

most important question gor boar or iit solve the integration (1/(3sinx+4cosx))

$${most}\:{important}\:{question}\:{gor}\:{boar}\:{or}\:{iit}\: \\ $$$${solve}\:{the}\:{integration}\:\:\frac{\mathrm{1}}{\mathrm{3}{sinx}+\mathrm{4}{cosx}} \\ $$

Question Number 28705    Answers: 1   Comments: 0

solve the integrayion ((sin2x)/(sin5xsin3x))

$${solve}\:{the}\:{integrayion}\:\frac{{sin}\mathrm{2}{x}}{{sin}\mathrm{5}{xsin}\mathrm{3}{x}} \\ $$

Question Number 28703    Answers: 0   Comments: 1

(1/((a^2 +x^2 )^(3/2) )) solve the integration

$$\frac{\mathrm{1}}{\left({a}^{\mathrm{2}} +{x}^{\mathrm{2}} \right)^{\mathrm{3}/\mathrm{2}} }\:\:\:{solve}\:{the}\:{integration} \\ $$

Question Number 28702    Answers: 0   Comments: 1

solve integration (1/(√((x−α)(β−x)))) .

$${solve}\:{integration}\:\:\frac{\mathrm{1}}{\sqrt{\left({x}−\alpha\right)\left(\beta−{x}\right)}}\:\:. \\ $$

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