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Question Number 37285    Answers: 0   Comments: 3

let A_n = ∫_0 ^∞ e^(−nx^2 ) sin((x/n))dx with n integr not 0 1) calculate A_n 2) find lim_(n→+∞) A_n

$${let}\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{nx}^{\mathrm{2}} } {sin}\left(\frac{{x}}{{n}}\right){dx}\:\:{with}\:{n}\:{integr}\:{not}\:\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} \\ $$

Question Number 37284    Answers: 0   Comments: 1

find A_n = ∫_0 ^1 (x^n /(ch(x))) dx .

$${find}\:\:{A}_{{n}} \:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{x}^{{n}} }{{ch}\left({x}\right)}\:{dx}\:. \\ $$

Question Number 37283    Answers: 0   Comments: 1

find ∫_0 ^∞ ((cosx)/(ch(x))) dx .

$${find}\:\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{cosx}}{{ch}\left({x}\right)}\:{dx}\:. \\ $$

Question Number 37282    Answers: 0   Comments: 5

let f(x)=(x/(1+x^2 +x^4 )) 1) find f^((n)) (x) 2)calculate f^((n)) (0) 3)developp f at integr serie.

$${let}\:{f}\left({x}\right)=\frac{{x}}{\mathrm{1}+{x}^{\mathrm{2}} \:+{x}^{\mathrm{4}} } \\ $$$$\left.\mathrm{1}\right)\:{find}\:{f}^{\left({n}\right)} \left({x}\right) \\ $$$$\left.\mathrm{2}\right){calculate}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{3}\right){developp}\:{f}\:{at}\:{integr}\:{serie}. \\ $$

Question Number 37281    Answers: 0   Comments: 1

find a better approximation for the integrals 1) ∫_0 ^1 e^(−x^2 ) dx 2) ∫_1 ^(+∞) e^(−x^2 ) dx .

$${find}\:{a}\:{better}\:{approximation}\:{for}\:{the} \\ $$$${integrals}\: \\ $$$$\left.\mathrm{1}\right)\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{e}^{−{x}^{\mathrm{2}} } {dx} \\ $$$$\left.\mathrm{2}\right)\:\int_{\mathrm{1}} ^{+\infty} \:{e}^{−{x}^{\mathrm{2}} } {dx}\:. \\ $$

Question Number 37280    Answers: 0   Comments: 1

calculate ∫_0 ^6 (e^(x−[x]) /(1+e^x ))dx .

$${calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{6}} \:\:\:\frac{{e}^{{x}−\left[{x}\right]} }{\mathrm{1}+{e}^{{x}} }{dx}\:. \\ $$

Question Number 37279    Answers: 1   Comments: 1

cslculate ∫∫_([0,1]^2 ) (x−y)e^(−x−y) dxdy .

$${cslculate}\:\int\int_{\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{2}} } \:\:\:\left({x}−{y}\right){e}^{−{x}−{y}} {dxdy}\:. \\ $$

Question Number 37278    Answers: 0   Comments: 1

calculate ∫∫_D x cos(x^2 +y^2 )dxdy with D={(x,y)∈R^2 / 0≤x≤1 and 1≤y≤3}

$$\:{calculate}\:\int\int_{{D}} \:{x}\:{cos}\left({x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \right){dxdy} \\ $$$${with}\:{D}=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /\:\mathrm{0}\leqslant{x}\leqslant\mathrm{1}\:{and}\right. \\ $$$$\left.\mathrm{1}\leqslant{y}\leqslant\mathrm{3}\right\} \\ $$

Question Number 37277    Answers: 0   Comments: 3

let f(x) = (1/(1+x^n )) with n integr 1)find f^′ (x) and f^(′′) (x) 2) find the poles of f 3)calculate f^((n)) (0) 4) developp f at integr serie.

$${let}\:{f}\left({x}\right)\:=\:\frac{\mathrm{1}}{\mathrm{1}+{x}^{{n}} }\:\:{with}\:{n}\:{integr} \\ $$$$\left.\mathrm{1}\right){find}\:{f}^{'} \left({x}\right)\:{and}\:{f}^{''} \left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{poles}\:{of}\:{f} \\ $$$$\left.\mathrm{3}\right){calculate}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{4}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie}. \\ $$

Question Number 37276    Answers: 0   Comments: 0

calculate I_n =∫_0 ^4 (−1)^([x]) (x^n −x)dx

$${calculate}\:\:{I}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{4}} \:\left(−\mathrm{1}\right)^{\left[{x}\right]} \left({x}^{{n}} \:−{x}\right){dx} \\ $$

Question Number 37275    Answers: 0   Comments: 0

let A_n = ∫_0 ^(1/n) arctan(1+x^2 )dx 1) calculate A_n 2)find lim_(n→+∞) A_n .

$${let}\:{A}_{{n}} \:=\:\int_{\mathrm{0}} ^{\frac{\mathrm{1}}{{n}}} \:{arctan}\left(\mathrm{1}+{x}^{\mathrm{2}} \right){dx} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}_{{n}} \\ $$$$\left.\mathrm{2}\right){find}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} \:. \\ $$

Question Number 37273    Answers: 0   Comments: 0

let f(x)=ln(x−sinx) 1)find D_f 2)developp f at integr serie.

$${let}\:{f}\left({x}\right)={ln}\left({x}−{sinx}\right) \\ $$$$\left.\mathrm{1}\right){find}\:{D}_{{f}} \\ $$$$\left.\mathrm{2}\right){developp}\:{f}\:{at}\:{integr}\:{serie}. \\ $$

Question Number 37272    Answers: 0   Comments: 0

let f(x)=cos(x−e^(−x) ) developp f at integr serie.

$${let}\:{f}\left({x}\right)={cos}\left({x}−{e}^{−{x}} \right) \\ $$$${developp}\:{f}\:{at}\:{integr}\:{serie}. \\ $$

Question Number 37271    Answers: 0   Comments: 2

find A_n =∫_1 ^2 ( 1 +(1/x) +(1/x^2 ) +...+(1/x^n ))^2 dx

$${find}\:\:{A}_{{n}} =\int_{\mathrm{1}} ^{\mathrm{2}} \left(\:\mathrm{1}\:+\frac{\mathrm{1}}{{x}}\:+\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\:+...+\frac{\mathrm{1}}{{x}^{{n}} }\right)^{\mathrm{2}} {dx} \\ $$

Question Number 37270    Answers: 1   Comments: 0

find ∫_0 ^1 (((1−x^(n+1) )/(1−x)))^2 dx .

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

Question Number 37269    Answers: 0   Comments: 2

let f(x)= e^(−2x) ln(1+x) developp f at integr serie .

$${let}\:{f}\left({x}\right)=\:{e}^{−\mathrm{2}{x}} {ln}\left(\mathrm{1}+{x}\right) \\ $$$${developp}\:{f}\:{at}\:{integr}\:{serie}\:. \\ $$

Question Number 37263    Answers: 1   Comments: 1

Question Number 37258    Answers: 1   Comments: 0

∫ ((x^3 +1)/(√(x^2 +x))) dx = ?

$$\int\:\frac{{x}^{\mathrm{3}} +\mathrm{1}}{\sqrt{{x}^{\mathrm{2}} +{x}}}\:{dx}\:=\:? \\ $$

Question Number 37252    Answers: 1   Comments: 1

Question Number 37540    Answers: 1   Comments: 0

For x>1 , ∫ sin^(−1) (((2x)/(1+x^2 )))dx = ?

$$\mathrm{For}\:{x}>\mathrm{1}\:,\: \\ $$$$\int\:\mathrm{sin}^{−\mathrm{1}} \left(\frac{\mathrm{2}{x}}{\mathrm{1}+{x}^{\mathrm{2}} }\right){dx}\:=\:? \\ $$

Question Number 37249    Answers: 0   Comments: 0

a triangle with vertices A(2,1),B(6,1) and (3,3) is transformed by (((2 0)),((0 2)) ) a)find the image A′B′C′ after this transformstion b) state the type of transformation

$${a}\:{triangle}\:{with}\:{vertices}\: \\ $$$$\:{A}\left(\mathrm{2},\mathrm{1}\right),{B}\left(\mathrm{6},\mathrm{1}\right)\:{and}\:\left(\mathrm{3},\mathrm{3}\right)\:{is}\:{transformed} \\ $$$${by}\:\begin{pmatrix}{\mathrm{2}\:\:\:\:\:\:\:\:\:\mathrm{0}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\mathrm{2}}\end{pmatrix}\: \\ $$$$\left.{a}\right){find}\:{the}\:{image}\:{A}'{B}'{C}'\:{after}\: \\ $$$${this}\:{transformstion} \\ $$$$\left.{b}\right)\:{state}\:{the}\:{type}\:{of}\:{transformation} \\ $$

Question Number 37244    Answers: 1   Comments: 0

sin^2 (π/11)+sin^2 (2π/11)+...+sin^2 (5π/11)=?

$$\mathrm{sin}^{\mathrm{2}} \left(\pi/\mathrm{11}\right)+\mathrm{sin}^{\mathrm{2}} \left(\mathrm{2}\pi/\mathrm{11}\right)+...+\mathrm{sin}^{\mathrm{2}} \left(\mathrm{5}\pi/\mathrm{11}\right)=? \\ $$

Question Number 37243    Answers: 0   Comments: 0

find f (t) =∫_0 ^∞ e^x ln(1+e^(−tx) )dx with t >0 . 2) let u_n = ∫_0 ^∞ e^x ln(1+e^(−nx) ) dx find lim_(n→+∞) u_n .

$${find}\:{f}\:\left({t}\right)\:=\int_{\mathrm{0}} ^{\infty} \:{e}^{{x}} {ln}\left(\mathrm{1}+{e}^{−{tx}} \right){dx}\:{with}\:{t}\:>\mathrm{0}\:. \\ $$$$\left.\mathrm{2}\right)\:{let}\:{u}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{{x}} {ln}\left(\mathrm{1}+{e}^{−{nx}} \right)\:{dx} \\ $$$${find}\:{lim}_{{n}\rightarrow+\infty} {u}_{{n}} \:. \\ $$

Question Number 37237    Answers: 1   Comments: 1

calculate ∫_0 ^(π/2) ((cosθ.sinθ)/(cosθ +sinθ)) dθ .

$${calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\frac{{cos}\theta.{sin}\theta}{{cos}\theta\:+{sin}\theta}\:{d}\theta\:. \\ $$

Question Number 37236    Answers: 0   Comments: 0

calculate ∫_0 ^(2π) ∣sin(((kt)/2))∣ dt with k integr and k≥3

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \mid{sin}\left(\frac{{kt}}{\mathrm{2}}\right)\mid\:{dt}\:\:{with}\:{k}\:{integr} \\ $$$${and}\:{k}\geqslant\mathrm{3} \\ $$

Question Number 37235    Answers: 1   Comments: 0

calculate ∫_0 ^(π/2) (√(4sin^2 t +cos^2 (t))) dt

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \sqrt{\mathrm{4}{sin}^{\mathrm{2}} {t}\:+{cos}^{\mathrm{2}} \left({t}\right)}\:\:{dt} \\ $$

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