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f and g are 2 function C^n on [a,b] prove that ∫_a ^b f^((n)) (x)g(x)dx=[Σ_(k=0) ^(n−1) (−1)^k f^((k)) g^((n−k)) ]_a ^b +(−1)^n ∫_a ^b f(x)g^((n)) (x)dx

$${f}\:{and}\:{g}\:{are}\:\mathrm{2}\:{function}\:\:{C}^{{n}} \:{on}\:\left[{a},{b}\right]\:{prove}\:{that} \\ $$$$\int_{{a}} ^{{b}} \:{f}^{\left({n}\right)} \left({x}\right){g}\left({x}\right){dx}=\left[\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \left(−\mathrm{1}\right)^{{k}} \:{f}^{\left({k}\right)} {g}^{\left({n}−{k}\right)} \right]_{{a}} ^{{b}} \:+\left(−\mathrm{1}\right)^{{n}} \int_{{a}} ^{{b}} {f}\left({x}\right){g}^{\left({n}\right)} \left({x}\right){dx} \\ $$

Question Number 30563 Answers: 0 Comments: 0

let f(x)=∣x−2 [((x+1)/2)]∣ 1) prove that f is periodic 2) simplify f(x) if p≤x+1 and p∈Z .

$${let}\:{f}\left({x}\right)=\mid{x}−\mathrm{2}\:\left[\frac{{x}+\mathrm{1}}{\mathrm{2}}\right]\mid \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{f}\:{is}\:{periodic} \\ $$$$\left.\mathrm{2}\right)\:{simplify}\:{f}\left({x}\right)\:{if}\:{p}\leqslant{x}+\mathrm{1}\:{and}\:{p}\in{Z}\:. \\ $$

Question Number 30560 Answers: 0 Comments: 0

study the roots of f_n (x)= Σ_(k=0) ^n (x^k /(k!)) .

$${study}\:{the}\:{roots}\:{of}\:{f}_{{n}} \left({x}\right)=\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\:\:\frac{{x}^{{k}} }{{k}!}\:. \\ $$

Question Number 30559 Answers: 0 Comments: 0

find I_n = ∫_0 ^1 (1−t^2 )^n dt .

$${find}\:\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\left(\mathrm{1}−{t}^{\mathrm{2}} \right)^{{n}} {dt}\:\:. \\ $$

Question Number 30558 Answers: 0 Comments: 0

find I= ∫_(1/2) ^1 arctan((√(1−x^2 )) dx .

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

Question Number 30557 Answers: 0 Comments: 0

if ϕ convexe and f continue on [a,b] prove that ϕ( (1/(b−a)) ∫_a ^b f(t)dt)≤ (1/(b−a)) ∫_a ^b ϕof(t)dt.

$${if}\:\varphi\:{convexe}\:{and}\:{f}\:{continue}\:{on}\:\left[{a},{b}\right]\:{prove}\:{that} \\ $$$$\varphi\left(\:\frac{\mathrm{1}}{{b}−{a}}\:\int_{{a}} ^{{b}} \:{f}\left({t}\right){dt}\right)\leqslant\:\frac{\mathrm{1}}{{b}−{a}}\:\int_{{a}} ^{{b}} \:\varphi{of}\left({t}\right){dt}. \\ $$

Question Number 30556 Answers: 0 Comments: 0

let S_n (x)= Σ_(k=1) ^n ((sin(kx))/(k^2 (k+1))) find lim_(n→∞) S_n (x).

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

Question Number 30555 Answers: 0 Comments: 0

find ∫_0 ^1 (dt/(√(1−t^4 ))) .

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dt}}{\sqrt{\mathrm{1}−{t}^{\mathrm{4}} }}\:. \\ $$

Question Number 30554 Answers: 0 Comments: 0

find ∫_0 ^∞ ((xcosθ +1)/(x^2 +2xcosθ +1))dx .

$${find}\:\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{xcos}\theta\:+\mathrm{1}}{{x}^{\mathrm{2}} \:+\mathrm{2}{xcos}\theta\:+\mathrm{1}}{dx}\:. \\ $$

Question Number 30553 Answers: 0 Comments: 3

find lim_(x→0) (1+sinx)^x −(1+x)^(sinx) .

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

Question Number 30552 Answers: 0 Comments: 0

find s(x)= Σ_(n≥0) ((sin(na))/((sina)^n )) (x^n /(n!)) and T(x) =Σ_(n≥0) ((cos(na))/((sina)^n )) (x^n /(n!)) .

$${find}\:{s}\left({x}\right)=\:\sum_{{n}\geqslant\mathrm{0}} \:\frac{{sin}\left({na}\right)}{\left({sina}\right)^{{n}} }\:\frac{{x}^{{n}} }{{n}!}\:{and}\: \\ $$$${T}\left({x}\right)\:=\sum_{{n}\geqslant\mathrm{0}} \:\:\frac{{cos}\left({na}\right)}{\left({sina}\right)^{{n}} }\:\frac{{x}^{{n}} }{{n}!}\:. \\ $$

Question Number 30551 Answers: 0 Comments: 0

find S = Σ_(n≥3) (1/((n+1)(n−2)2^n )) .

$$\:{find}\:\:{S}\:=\:\sum_{{n}\geqslant\mathrm{3}} \:\:\:\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)\left({n}−\mathrm{2}\right)\mathrm{2}^{{n}} }\:. \\ $$

Question Number 30550 Answers: 0 Comments: 0

let f(z)= Σ_(n≥0) a_n z^n /a_0 =1 ,a_1 =3 and ∀n≥2 a_n =3a_(n−1) −2 a_(n−2) find f(z) for ∣z∣<1 (z∈C) .

$${let}\:{f}\left({z}\right)=\:\sum_{{n}\geqslant\mathrm{0}} {a}_{{n}} {z}^{{n}} \:\:\:/{a}_{\mathrm{0}} =\mathrm{1}\:,{a}_{\mathrm{1}} =\mathrm{3}\:{and}\:\forall{n}\geqslant\mathrm{2} \\ $$$${a}_{{n}} =\mathrm{3}{a}_{{n}−\mathrm{1}} −\mathrm{2}\:{a}_{{n}−\mathrm{2}} \:\:\:\:{find}\:{f}\left({z}\right)\:{for}\:\mid{z}\mid<\mathrm{1}\:\:\left({z}\in{C}\right)\:. \\ $$

Question Number 30549 Answers: 0 Comments: 0

let S(x)= Σ_(n=0) ^∞ (x^(3n) /((3n)!)) find S(x).

$${let}\:{S}\left({x}\right)=\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{{x}^{\mathrm{3}{n}} }{\left(\mathrm{3}{n}\right)!}\:\:{find}\:{S}\left({x}\right). \\ $$

Question Number 30548 Answers: 0 Comments: 0

let put for ∣λ∣<1 u_n = ∫_0 ^π ((cos(nx))/(1−2λcosx +λ^2 ))dx find u_n interms of n and λ.

$${let}\:{put}\:\:{for}\:\mid\lambda\mid<\mathrm{1}\:\:\:\:{u}_{{n}} =\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\:\frac{{cos}\left({nx}\right)}{\mathrm{1}−\mathrm{2}\lambda{cosx}\:+\lambda^{\mathrm{2}} }{dx}\: \\ $$$${find}\:{u}_{{n}} \:{interms}\:{of}\:{n}\:{and}\:\lambda. \\ $$

Question Number 30547 Answers: 0 Comments: 0

ind S= Σ_(n=0) ^∞ ((n^3 +n^2 +n+1)/(n!)) .

$${ind}\:{S}=\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\frac{{n}^{\mathrm{3}} \:+{n}^{\mathrm{2}} \:+{n}+\mathrm{1}}{{n}!}\:\:. \\ $$

Question Number 30546 Answers: 0 Comments: 0

find I = ∫_1 ^(+∞) (((−1)^([x]) )/x^2 )dx .

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

Question Number 30545 Answers: 0 Comments: 0

Question Number 30544 Answers: 0 Comments: 0

find I= ∫_0 ^π (t/(2+sint))dt.

$${find}\:{I}=\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{{t}}{\mathrm{2}+{sint}}{dt}. \\ $$

Question Number 30543 Answers: 1 Comments: 1

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