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Question Number 35851 Answers: 1 Comments: 1

Question Number 35691 Answers: 0 Comments: 0

calculate lim_(a→0^+ ) ∫_(−a) ^a (√((1+x^2 )/(a^2 −x^2 ))) dx .

$${calculate}\:{lim}_{{a}\rightarrow\mathrm{0}^{+} \:\:\:\:} \:\:\:\int_{−{a}} ^{{a}} \:\:\sqrt{\frac{\mathrm{1}+{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} \:−{x}^{\mathrm{2}} }}\:\:{dx}\:. \\ $$

Question Number 35690 Answers: 0 Comments: 0

let B_n =Σ_(k=1) ^n sin(((kπ)/n)) sin((k/n^2 )) find lim_(n→+∞) B_n

$${let}\:{B}_{{n}} \:=\sum_{{k}=\mathrm{1}} ^{{n}} \:\:{sin}\left(\frac{{k}\pi}{{n}}\right)\:{sin}\left(\frac{{k}}{{n}^{\mathrm{2}} }\right) \\ $$$${find}\:{lim}_{{n}\rightarrow+\infty} \:\:{B}_{{n}} \\ $$

Question Number 35689 Answers: 1 Comments: 2

let S_n = Σ_(k=1) ^n (k^2 /(n^2 (√(n^2 +k^2 )))) find lim_(n→+∞) S_n

$${let}\:{S}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\:\frac{{k}^{\mathrm{2}} }{{n}^{\mathrm{2}} \sqrt{{n}^{\mathrm{2}} \:+{k}^{\mathrm{2}} }} \\ $$$${find}\:{lim}_{{n}\rightarrow+\infty} \:\:{S}_{{n}} \\ $$

Question Number 35688 Answers: 1 Comments: 1

let A_n =Σ_(k=1) ^n (1/(k+n))ln(1+(k/n)) calculate lim_(n→+∞) A_n

$${let}\:{A}_{{n}} \:=\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\:\frac{\mathrm{1}}{{k}+{n}}{ln}\left(\mathrm{1}+\frac{{k}}{{n}}\right) \\ $$$${calculate}\:{lim}_{{n}\rightarrow+\infty} {A}_{{n}} \\ $$

Question Number 35687 Answers: 1 Comments: 2

calculate f(a)=∫_0 ^π (dx/(1−a cosx)) a from R . 2) application calculate ∫_0 ^π (dx/(1−2cosx))

$${calculate}\:{f}\left({a}\right)=\int_{\mathrm{0}} ^{\pi} \:\:\:\:\:\frac{{dx}}{\mathrm{1}−{a}\:{cosx}}\:\:{a}\:{from}\:{R}\:. \\ $$$$\left.\mathrm{2}\right)\:{application}\:\:{calculate}\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{dx}}{\mathrm{1}−\mathrm{2}{cosx}} \\ $$

Question Number 35686 Answers: 1 Comments: 1

calculate ∫_(√3) ^(+∞) (dx/(x(√( 2+x^2 )))) .

$${calculate}\:\:\int_{\sqrt{\mathrm{3}}} ^{+\infty} \:\:\:\:\:\frac{{dx}}{{x}\sqrt{\:\mathrm{2}+{x}^{\mathrm{2}} }}\:. \\ $$

Question Number 35685 Answers: 1 Comments: 1

calculate ∫_0 ^(π/4) x artan(2x+1)dx

$${calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:{x}\:{artan}\left(\mathrm{2}{x}+\mathrm{1}\right){dx} \\ $$

Question Number 35684 Answers: 1 Comments: 1

calculate I = ∫_0 ^1 e^(2t) ln(1+e^t )dt

$${calculate}\:{I}\:\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{e}^{\mathrm{2}{t}} \:{ln}\left(\mathrm{1}+{e}^{{t}} \right){dt} \\ $$

Question Number 35683 Answers: 1 Comments: 1

find ∫ x^2 ln(x^6 −1)dx

$${find}\:\int\:\:{x}^{\mathrm{2}} {ln}\left({x}^{\mathrm{6}} −\mathrm{1}\right){dx} \\ $$

Question Number 35682 Answers: 1 Comments: 2

let F(x) = ∫_(x +1) ^(x^2 +1) arctan(1+t)dt 1) calculate (∂F/∂x)(x) 2) find lim_(x→0) F(x) .

$${let}\:{F}\left({x}\right)\:=\:\int_{{x}\:+\mathrm{1}} ^{{x}^{\mathrm{2}} \:+\mathrm{1}} \:\:\:{arctan}\left(\mathrm{1}+{t}\right){dt} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\frac{\partial{F}}{\partial{x}}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:\:{find}\:{lim}_{{x}\rightarrow\mathrm{0}} \:{F}\left({x}\right)\:. \\ $$

Question Number 35681 Answers: 1 Comments: 1

find ∫ arctan(x)dx

$${find}\:\:\int\:{arctan}\left({x}\right){dx} \\ $$

Question Number 35680 Answers: 0 Comments: 0

by using residus theorem calculate W_n =∫_0 ^(π/2) cos^(2n) t dt ( wallis integal) n integr natural .

$${by}\:{using}\:{residus}\:{theorem}\:{calculate} \\ $$$${W}_{{n}} \:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:{cos}^{\mathrm{2}{n}} {t}\:{dt}\:\:\left(\:\:{wallis}\:{integal}\right)\:{n}\:{integr} \\ $$$${natural}\:. \\ $$

Question Number 35678 Answers: 0 Comments: 1

let f(t) =∫_0 ^∞ ((e^(−tx^2 ) arctan(x^2 ))/x^2 )dx with t>0 1) study the existencte of f(t) 2)calculate f^′ (t) 3)find a simple form of f(t).

$${let}\:{f}\left({t}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{tx}^{\mathrm{2}} } \:{arctan}\left({x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} }{dx}\:{with}\:{t}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{study}\:{the}\:{existencte}\:{of}\:{f}\left({t}\right) \\ $$$$\left.\mathrm{2}\right){calculate}\:{f}^{'} \left({t}\right) \\ $$$$\left.\mathrm{3}\right){find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({t}\right). \\ $$

Question Number 35677 Answers: 0 Comments: 2

find F(x)=∫_0 ^x e^(−2t) cos(t+(π/4))dx.

$${find}\:{F}\left({x}\right)=\int_{\mathrm{0}} ^{{x}} \:{e}^{−\mathrm{2}{t}} {cos}\left({t}+\frac{\pi}{\mathrm{4}}\right){dx}. \\ $$

Question Number 35676 Answers: 0 Comments: 1

find f(x)=∫_0 ^x ch^4 t dt

$${find}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{{x}} \:{ch}^{\mathrm{4}} {t}\:{dt} \\ $$

Question Number 35675 Answers: 0 Comments: 1

calculate ∫_1 ^3 (x/(e^x −1))dx ..

$${calculate}\:\:\int_{\mathrm{1}} ^{\mathrm{3}} \:\:\:\frac{{x}}{{e}^{{x}} \:−\mathrm{1}}{dx}\:.. \\ $$

Question Number 35656 Answers: 0 Comments: 5

Following alphabet lacks one letter. abcdefghijklmnopqrstuvxyz I request that letter, please come and make the alphabet complete.

$$\mathrm{Following}\:\mathrm{alphabet}\:\mathrm{lacks}\:\mathrm{one}\:\mathrm{letter}. \\ $$$$\:\:\:\:\:\mathrm{abcdefghijklmnopqrstuvxyz} \\ $$$$\mathrm{I}\:\mathrm{request}\:\mathrm{that}\:\mathrm{letter}, \\ $$$$\mathrm{please}\:\mathrm{come}\:\mathrm{and}\:\mathrm{make}\:\mathrm{the}\:\mathrm{alphabet} \\ $$$$\mathrm{complete}. \\ $$$$ \\ $$

Question Number 35654 Answers: 1 Comments: 0

if cos^2 θ−sin^2 θ=tan^2 ∅ Then proof that 2cos^2 ∅−1=cos^2 ∅−sin^2 ∅=2tan^2 θ

$${if}\:\:\mathrm{cos}\:^{\mathrm{2}} \theta−\mathrm{sin}\:^{\mathrm{2}} \theta=\mathrm{tan}\:^{\mathrm{2}} \emptyset\:\:{Then}\:{proof}\:{that} \\ $$$$\mathrm{2cos}\:^{\mathrm{2}} \emptyset−\mathrm{1}=\mathrm{cos}\:^{\mathrm{2}} \emptyset−\mathrm{sin}\:^{\mathrm{2}} \emptyset=\mathrm{2tan}\:^{\mathrm{2}} \theta \\ $$

Question Number 35642 Answers: 1 Comments: 2

If y= (√((a−x)(x−b)))−(a−b)tan^(−1) ((((a−x)/(x−b)))^(0.5) ). Then find (dy/dx) ?

$${If}\:{y}=\:\sqrt{\left({a}−{x}\right)\left({x}−{b}\right)}−\left({a}−{b}\right)\mathrm{tan}^{−\mathrm{1}} \left(\left(\frac{{a}−{x}}{{x}−{b}}\right)^{\mathrm{0}.\mathrm{5}} \right). \\ $$$${Then}\:{find}\:\frac{{dy}}{{dx}}\:? \\ $$

Question Number 35640 Answers: 1 Comments: 0

A panel of 3 women and 4 men is to be formed from 8 women and 7 men.Find the number of ways which the panel can be formed if it must contain at least 2 women.

$${A}\:{panel}\:{of}\:\mathrm{3}\:{women}\:{and}\:\mathrm{4}\:{men}\:{is} \\ $$$${to}\:{be}\:{formed}\:{from}\:\mathrm{8}\:{women}\:{and} \\ $$$$\mathrm{7}\:{men}.{Find}\:{the}\:{number}\:{of}\:{ways} \\ $$$${which}\:{the}\:{panel}\:{can}\:{be}\:{formed}\:{if} \\ $$$${it}\:{must}\:{contain}\:{at}\:{least}\:\mathrm{2}\:{women}. \\ $$

Question Number 35639 Answers: 0 Comments: 1

Three boys,two girls and a puppy sit at a round table.In how many ways can they be arranged if the puppy is to be seated i)between the two girls ii)between any two boys

$${Three}\:{boys},{two}\:{girls}\:{and}\:{a}\:{puppy} \\ $$$${sit}\:{at}\:{a}\:{round}\:{table}.{In}\:{how}\:{many} \\ $$$${ways}\:{can}\:{they}\:{be}\:{arranged}\:{if}\:{the} \\ $$$${puppy}\:{is}\:{to}\:{be}\:{seated} \\ $$$$\left.{i}\right){between}\:{the}\:{two}\:{girls} \\ $$$$\left.{ii}\right){between}\:{any}\:{two}\:{boys} \\ $$

Question Number 35635 Answers: 1 Comments: 1

Question Number 35992 Answers: 0 Comments: 1

let f(x)= ((sin(2x))/x) χ_(]−a,a[) (x) with a>0 calculate the fourier trsnsform of f .

$${let}\:{f}\left({x}\right)=\:\frac{{sin}\left(\mathrm{2}{x}\right)}{{x}}\:\chi_{\left.\right]−{a},{a}\left[\right.} \left({x}\right)\:\:{with}\:{a}>\mathrm{0} \\ $$$${calculate}\:{the}\:{fourier}\:{trsnsform}\:{of}\:{f}\:. \\ $$

Question Number 35632 Answers: 0 Comments: 2

let ϕ(x)= (1/(√(a^2 −x^2 ))) if ∣x∣<a and ϕ(x)=0 if ∣x∣≥a find the fourier transform of ϕ .

$${let}\:\varphi\left({x}\right)=\:\frac{\mathrm{1}}{\sqrt{{a}^{\mathrm{2}} −{x}^{\mathrm{2}} }}\:\:{if}\:\mid{x}\mid<{a}\:\:{and}\:\varphi\left({x}\right)=\mathrm{0}\:{if}\:\mid{x}\mid\geqslant{a} \\ $$$${find}\:{the}\:{fourier}\:{transform}\:{of}\:\varphi\:. \\ $$

Question Number 35631 Answers: 0 Comments: 0

let U_n = ∫_0 ^∞ e^(−(t/n)) arctan(t)dt find a equivalent of U_n (n→+∞)

$${let}\:{U}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:\:\:{e}^{−\frac{{t}}{{n}}} \:\:{arctan}\left({t}\right){dt} \\ $$$${find}\:{a}\:{equivalent}\:{of}\:{U}_{{n}} \:\:\left({n}\rightarrow+\infty\right) \\ $$

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