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AllQuestion and Answers: Page 1765

Question Number 30517 Answers: 0 Comments: 1

let g(x)= e^x cosx find g^((n)) (x) .

$${let}\:{g}\left({x}\right)=\:{e}^{{x}} {cosx}\:\:{find}\:\:{g}^{\left({n}\right)} \left({x}\right)\:. \\ $$

Question Number 30515 Answers: 0 Comments: 0

let f(x)= (1/(√(1+x^2 ))) find a form of f^((n)) (x) .

$${let}\:{f}\left({x}\right)=\:\frac{\mathrm{1}}{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:{find}\:{a}\:{form}\:{of}\:{f}^{\left({n}\right)} \left({x}\right)\:. \\ $$

Question Number 30514 Answers: 0 Comments: 0

find lim_(n→∞) Π_(k=1) ^n (1− (k^2 /n^3 )).

$${find}\:{lim}_{{n}\rightarrow\infty} \:\prod_{{k}=\mathrm{1}} ^{{n}} \:\left(\mathrm{1}−\:\frac{{k}^{\mathrm{2}} }{{n}^{\mathrm{3}} }\right).\: \\ $$

Question Number 30513 Answers: 0 Comments: 1

Question Number 30512 Answers: 0 Comments: 1

find I =∫_0 ^1 (√((1−t)/(1+t))) dt .

$${find}\:\:{I}\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\sqrt{\frac{\mathrm{1}−{t}}{\mathrm{1}+{t}}}\:{dt}\:. \\ $$

Question Number 30511 Answers: 0 Comments: 1

find lim_(x→+∞) e^x [(1/x)].

$${find}\:\:{lim}_{{x}\rightarrow+\infty} \:\:{e}^{{x}} \:\left[\frac{\mathrm{1}}{{x}}\right]. \\ $$

Question Number 30510 Answers: 0 Comments: 0

find lim_(x→0^(+ ) ) (√x) [ (1/x)] and lim_(x→+∞) (([x])/x) .

$${find}\:\:{lim}_{{x}\rightarrow\mathrm{0}^{+\:} } \:\:\:\:\sqrt{{x}}\:\left[\:\frac{\mathrm{1}}{{x}}\right]\:\:{and}\:{lim}_{{x}\rightarrow+\infty} \:\:\frac{\left[{x}\right]}{{x}}\:. \\ $$

Question Number 30508 Answers: 0 Comments: 1

find I= ∫ e^(arcsinx) dx .

$${find}\:{I}=\:\int\:\:{e}^{{arcsinx}} {dx}\:. \\ $$

Question Number 30507 Answers: 0 Comments: 0

find ∫_(−π) ^π (dx/(2+cosx)) 2) if (1/(2+cosx))= (a_0 /2) +Σ_(n≥1) a_n cos(nx) find a_0 and a_n .

$${find}\:\int_{−\pi} ^{\pi} \:\:\frac{{dx}}{\mathrm{2}+{cosx}} \\ $$$$\left.\mathrm{2}\right)\:{if}\:\:\frac{\mathrm{1}}{\mathrm{2}+{cosx}}=\:\frac{{a}_{\mathrm{0}} }{\mathrm{2}}\:+\sum_{{n}\geqslant\mathrm{1}} {a}_{{n}} \:{cos}\left({nx}\right)\:\:{find}\:{a}_{\mathrm{0}} \:{and}\:{a}_{{n}} \:. \\ $$

Question Number 30506 Answers: 0 Comments: 0

find f(x) =∫_0 ^x (t/(1+t^4 ))dt with x>0.

$${find}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{{x}} \:\:\:\frac{{t}}{\mathrm{1}+{t}^{\mathrm{4}} }{dt}\:{with}\:{x}>\mathrm{0}. \\ $$

Question Number 30509 Answers: 0 Comments: 0

f function continue at o and lim_(x→0) ((f(2x)−f(x))/x)=l prove that f is derivable at o and f^′ (0)=l.

$${f}\:{function}\:{continue}\:{at}\:{o}\:{and}\:{lim}_{{x}\rightarrow\mathrm{0}} \frac{{f}\left(\mathrm{2}{x}\right)−{f}\left({x}\right)}{{x}}={l} \\ $$$${prove}\:{that}\:{f}\:{is}\:{derivable}\:{at}\:{o}\:{and}\:{f}^{'} \left(\mathrm{0}\right)={l}. \\ $$

Question Number 30505 Answers: 0 Comments: 0

find A=Σ_(k=0) ^n ch(a+kb) and B=Σ_(k=0) ^n sh(a+kb).

$${find}\:\:{A}=\sum_{{k}=\mathrm{0}} ^{{n}} \:{ch}\left({a}+{kb}\right)\:{and}\:{B}=\sum_{{k}=\mathrm{0}} ^{{n}} \:{sh}\left({a}+{kb}\right). \\ $$

Question Number 30504 Answers: 1 Comments: 0

find lim_(x→∞) x^2 ( e^(1/x) − e^(1/(x+1)) ) .

$${find}\:{lim}_{{x}\rightarrow\infty} \:{x}^{\mathrm{2}} \left(\:{e}^{\frac{\mathrm{1}}{{x}}} \:\:\:−\:{e}^{\frac{\mathrm{1}}{{x}+\mathrm{1}}} \right)\:. \\ $$

Question Number 30502 Answers: 1 Comments: 0

find lim_(x→0) (sinx +cosx)^(1/x) .

$${find}\:\:{lim}_{{x}\rightarrow\mathrm{0}} \left({sinx}\:+{cosx}\right)^{\frac{\mathrm{1}}{{x}}} \:\:. \\ $$

Question Number 30501 Answers: 0 Comments: 0

let put w=e^(i((2π)/n)) find Σ_(k=1) ^n (x+w^k )^n 2) find Σ_(k=1) ^n n(x+w^k )^(n−1) .

$${let}\:{put}\:{w}={e}^{{i}\frac{\mathrm{2}\pi}{{n}}} \:\:\:\:\:{find}\:\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\left({x}+{w}^{{k}} \right)^{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\:\sum_{{k}=\mathrm{1}} ^{{n}} {n}\left({x}+{w}^{{k}} \right)^{{n}−\mathrm{1}} \:\:. \\ $$$$ \\ $$

Question Number 30500 Answers: 0 Comments: 0

find ∫_1 ^(+∞) (dt/(t^2 (1+t^2 ))) .

$${find}\:\int_{\mathrm{1}} ^{+\infty} \:\:\:\frac{{dt}}{{t}^{\mathrm{2}} \left(\mathrm{1}+{t}^{\mathrm{2}} \right)}\:. \\ $$

Question Number 30499 Answers: 0 Comments: 0

let put F(x)= ∫_0 ^x (√(tant)) dt with x>0 find F(x).

$${let}\:{put}\:{F}\left({x}\right)=\:\int_{\mathrm{0}} ^{{x}} \:\sqrt{{tant}}\:\:{dt}\:{with}\:{x}>\mathrm{0}\:\:{find}\:{F}\left({x}\right). \\ $$

Question Number 30498 Answers: 1 Comments: 0

find I= ∫_0 ^(√3) arcsin(((2x)/(1+x^2 )))dx .

$${find}\:\:{I}=\:\int_{\mathrm{0}} ^{\sqrt{\mathrm{3}}} \:{arcsin}\left(\frac{\mathrm{2}{x}}{\mathrm{1}+{x}^{\mathrm{2}} }\right){dx}\:\:. \\ $$

Question Number 30497 Answers: 0 Comments: 0

integrate 2xy^′ −y =(2/3) x^(3/2) .

$${integrate}\:\:\mathrm{2}{xy}^{'} \:−{y}\:=\frac{\mathrm{2}}{\mathrm{3}}\:{x}^{\frac{\mathrm{3}}{\mathrm{2}}} \:. \\ $$

Question Number 30496 Answers: 0 Comments: 0

find A_n = Σ_(k=0) ^n C_n ^k cos(kx) and B_n =Σ_(k=0) ^n C_n ^k sin(kx)

$${find}\:\:{A}_{{n}} =\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \:{cos}\left({kx}\right)\:{and}\:{B}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \:{sin}\left({kx}\right) \\ $$

Question Number 30495 Answers: 0 Comments: 0

let f(x)=(√(1+x^2 )) find f^((n)) (x) and calculate f^((n)) (0).

$${let}\:{f}\left({x}\right)=\sqrt{\mathrm{1}+{x}^{\mathrm{2}} \:}\:\:\:\:{find}\:{f}^{\left({n}\right)} \left({x}\right)\:{and}\:{calculate}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right). \\ $$

Question Number 30494 Answers: 1 Comments: 0

find I= ∫_0 ^1 (dx/((1+x)(√(1+x^2 )))) .

$${find}\:{I}=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}\right)\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:\:. \\ $$

Question Number 30493 Answers: 0 Comments: 0

study tbe sequence x_(n+1) = (1/(2−x_n )) with x_o ≠2.

$${study}\:{tbe}\:{sequence}\:\:{x}_{{n}+\mathrm{1}} \:=\:\frac{\mathrm{1}}{\mathrm{2}−{x}_{{n}} }\:{with}\:{x}_{{o}} \neq\mathrm{2}. \\ $$

Question Number 30492 Answers: 0 Comments: 0

let (u_(n)) / u_(n+1) = u_n +(1/n) find a equivalent of u_n for n→∞ .

$${let}\:\left({u}_{\left.{n}\right)} \:\:\:/\:\:\:{u}_{{n}+\mathrm{1}} =\:{u}_{{n}} \:\:+\frac{\mathrm{1}}{{n}}\:\:\:{find}\:{a}\:{equivalent}\:{of}\:{u}_{{n}} \:{for}\right. \\ $$$${n}\rightarrow\infty\:. \\ $$$$ \\ $$

Question Number 30491 Answers: 0 Comments: 0

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

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

Question Number 30490 Answers: 0 Comments: 0

f function derivable at o and f(0)=0 let S_n = Σ_(k=0) ^n f((k/n^2 )) .find lim_(n→∞) S_n .

$${f}\:{function}\:{derivable}\:{at}\:{o}\:{and}\:{f}\left(\mathrm{0}\right)=\mathrm{0}\:{let} \\ $$$${S}_{{n}} =\:\sum_{{k}=\mathrm{0}} ^{{n}} {f}\left(\frac{{k}}{{n}^{\mathrm{2}} }\right)\:\:.{find}\:{lim}_{{n}\rightarrow\infty} {S}_{{n}} . \\ $$

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