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AllQuestion and Answers: Page 1778

Question Number 30523 Answers: 0 Comments: 0

(α_k )_(0≤k≤n−1) are roots of x^n −1 simplify Π_n =Π_(k=0) ^(n−1) (x+α_k y) .

$$\:\left(\alpha_{{k}} \right)_{\mathrm{0}\leqslant{k}\leqslant{n}−\mathrm{1}} {are}\:{roots}\:{of}\:\:{x}^{{n}} −\mathrm{1}\:\:{simplify} \\ $$$$\prod_{{n}} =\prod_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:\left({x}+\alpha_{{k}} {y}\right)\:. \\ $$

Question Number 30522 Answers: 1 Comments: 1

let p(x)= (1+ix)^n −(1−ix)^n 1) find the roots of p(x) and factorize p(x) ) give p(x) at form of arcs.

$${let}\:{p}\left({x}\right)=\:\left(\mathrm{1}+{ix}\right)^{{n}} \:−\left(\mathrm{1}−{ix}\right)^{{n}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{roots}\:{of}\:{p}\left({x}\right)\:{and}\:{factorize}\:{p}\left({x}\right) \\ $$$$\left.\right)\:{give}\:{p}\left({x}\right)\:{at}\:{form}\:{of}\:{arcs}. \\ $$$$ \\ $$

Question Number 30521 Answers: 0 Comments: 2

1) find ∫_0 ^1 ((√(1+x^2 )))^n cos(narctanx)dx 2)find ∫_0 ^1 ((√(1+x^2 )))^3 cos(3 arctanx)dx .

$$\left.\mathrm{1}\right)\:{find}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\left(\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\right)^{{n}} \:{cos}\left({narctanx}\right){dx} \\ $$$$\left.\mathrm{2}\right){find}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \left(\sqrt{\mathrm{1}+{x}^{\mathrm{2}} \:}\right)^{\mathrm{3}} \:{cos}\left(\mathrm{3}\:{arctanx}\right){dx}\:. \\ $$

Question Number 30519 Answers: 1 Comments: 0

let j=e^(i((2π)/3)) find the value of Σ_(k=0) ^n C_n ^k (1+j)^k j^(2n−2k) .

$${let}\:{j}={e}^{{i}\frac{\mathrm{2}\pi}{\mathrm{3}}} \:\:\:\:{find}\:{the}\:{value}\:{of}\:\sum_{{k}=\mathrm{0}} ^{{n}} {C}_{{n}} ^{{k}} \left(\mathrm{1}+{j}\right)^{{k}} {j}^{\mathrm{2}{n}−\mathrm{2}{k}} \:. \\ $$

Question Number 30518 Answers: 1 Comments: 0

let a>0 find f(a) =∫_0 ^∞ (dx/((x+a)(√(a^2 +x^2 )))) .

$${let}\:{a}>\mathrm{0}\:{find}\:\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{dx}}{\left({x}+{a}\right)\sqrt{{a}^{\mathrm{2}} \:+{x}^{\mathrm{2}} }}\:. \\ $$

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).

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

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