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AllQuestion and Answers: Page 1702

Question Number 32361 Answers: 0 Comments: 1

let give a>0 find ∫_0 ^∞ (e^(−x) /(√(x+a))) dx.

$${let}\:{give}\:{a}>\mathrm{0}\:{find}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{e}^{−{x}} }{\sqrt{{x}+{a}}}\:{dx}. \\ $$

Question Number 32360 Answers: 0 Comments: 1

find the value of ∫_0 ^∞ (dx/((2x+1)(2x+3))) .

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\left(\mathrm{2}{x}+\mathrm{1}\right)\left(\mathrm{2}{x}+\mathrm{3}\right)}\:. \\ $$

Question Number 32359 Answers: 0 Comments: 1

find ∫_0 ^∞ (dx/((1+x^2 )(1+x^4 ))) .

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{x}^{\mathrm{4}} \right)}\:. \\ $$

Question Number 32357 Answers: 1 Comments: 0

x^2 +ax−24=0 root is integer a range i cant speak english well. sorry

$${x}^{\mathrm{2}} +{ax}−\mathrm{24}=\mathrm{0} \\ $$$${root}\:{is}\:{integer} \\ $$$${a}\:\:\:{range} \\ $$$$ \\ $$$${i}\:{cant}\:{speak}\:{english}\:{well}.\:{sorry} \\ $$

Question Number 32354 Answers: 0 Comments: 1

calculate ∫_0 ^(π/4) (dt/((1+sin^2 t)^2 )) .

$${calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\:\:\:\:\frac{{dt}}{\left(\mathrm{1}+{sin}^{\mathrm{2}} {t}\right)^{\mathrm{2}} }\:. \\ $$

Question Number 32353 Answers: 1 Comments: 0

calculate ∫_0 ^(π/4) cos(x)ln(cos(x))dx .

$${calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:{cos}\left({x}\right){ln}\left({cos}\left({x}\right)\right){dx}\:. \\ $$

Question Number 32352 Answers: 1 Comments: 2

find the value of ∫_0 ^1 arctan((√(1−x^2 )))dx

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{arctan}\left(\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\right){dx} \\ $$

Question Number 32351 Answers: 1 Comments: 0

calculate ∫_0 ^(π/2) (dt/(1+cosθ sint)) .

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\:\:\frac{{dt}}{\mathrm{1}+{cos}\theta\:{sint}}\:.\: \\ $$

Question Number 32350 Answers: 0 Comments: 0

calculate ∫_0 ^1 ^3 (√(x^2 (1−x))) dx

$${calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:^{\mathrm{3}} \sqrt{{x}^{\mathrm{2}} \left(\mathrm{1}−{x}\right)}\:{dx} \\ $$

Question Number 32349 Answers: 0 Comments: 1

find the value of ∫_0 ^π ((xdx)/(1+sinx)) .

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{xdx}}{\mathrm{1}+{sinx}}\:. \\ $$

Question Number 32348 Answers: 0 Comments: 0

1)let n ∈Nand A_n = ∫_0 ^π (dx/(1+cos^2 (nx))) .calculate A_n 2) f∈ C^0 ([0,π], R) find lim_(n→∞) ∫_0 ^π ((f(x))/(1+cos^2 (nx)))dx .

$$\left.\mathrm{1}\right){let}\:{n}\:\in{Nand}\:\:\:{A}_{{n}} \:=\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{dx}}{\mathrm{1}+{cos}^{\mathrm{2}} \left({nx}\right)}\:.{calculate}\:{A}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{f}\in\:{C}^{\mathrm{0}} \left(\left[\mathrm{0},\pi\right],\:{R}\right)\:{find}\:{lim}_{{n}\rightarrow\infty} \:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{f}\left({x}\right)}{\mathrm{1}+{cos}^{\mathrm{2}} \left({nx}\right)}{dx}\:. \\ $$

Question Number 32346 Answers: 0 Comments: 0

let u_0 =1 and u_(n+1) = u_n ((1+2u_n )/(1+3n)) give a equivalent of u_(n )

$${let}\:{u}_{\mathrm{0}} =\mathrm{1}\:{and}\:\:{u}_{{n}+\mathrm{1}} =\:{u}_{{n}} \:\frac{\mathrm{1}+\mathrm{2}{u}_{{n}} }{\mathrm{1}+\mathrm{3}{n}} \\ $$$${give}\:{a}\:{equivalent}\:{of}\:{u}_{{n}\:} \\ $$

Question Number 32345 Answers: 0 Comments: 0

calculate lim_(n→∞) Σ_(i=1) ^n Σ_(j=1) ^n (((−1)^(i+j) )/(i+j)) .

$${calculate}\:{lim}_{{n}\rightarrow\infty} \:\:\sum_{{i}=\mathrm{1}} ^{{n}} \:\sum_{{j}=\mathrm{1}} ^{{n}} \:\:\:\frac{\left(−\mathrm{1}\right)^{{i}+{j}} }{{i}+{j}}\:. \\ $$

Question Number 32344 Answers: 0 Comments: 0

let u_n = Σ_(k=1) ^n ch((1/(√(k+n)))) −n prove that u_n is convergent and find its limit.

$${let}\:\:{u}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{ch}\left(\frac{\mathrm{1}}{\sqrt{{k}+{n}}}\right)\:−{n} \\ $$$${prove}\:{that}\:{u}_{{n}} \:{is}\:{convergent}\:{and}\:{find}\:{its}\:{limit}. \\ $$

Question Number 32343 Answers: 1 Comments: 1

calculate ∫_0 ^(2π) (dt/(x−e^(it) )) .

$${calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\:\frac{{dt}}{{x}−{e}^{{it}} }\:\:. \\ $$

Question Number 32342 Answers: 0 Comments: 0

find the value of ∫∫_D ((dxdy)/((4x^2 +y^2 +1)^2 )) D={(x,y)∈ R^2 / x^2 +y^2 ≤1 and y ≤2x } .

$${find}\:{the}\:{value}\:{of}\:\int\int_{{D}} \:\:\:\frac{{dxdy}}{\left(\mathrm{4}{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} } \\ $$$${D}=\left\{\left({x},{y}\right)\in\:{R}^{\mathrm{2}} \:/\:{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:\leqslant\mathrm{1}\:{and}\:{y}\:\leqslant\mathrm{2}{x}\:\right\}\:. \\ $$

Question Number 32341 Answers: 0 Comments: 1

let give λ from R and λ^2 ≠1 and I_n (λ) = ∫_0 ^π ((cos(nt))/(1−2λcost +λ^2 ))dt .calculate I_n (λ).

$${let}\:{give}\:\lambda\:{from}\:{R}\:{and}\:\lambda^{\mathrm{2}} \neq\mathrm{1}\:{and} \\ $$$${I}_{{n}} \left(\lambda\right)\:=\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{cos}\left({nt}\right)}{\mathrm{1}−\mathrm{2}\lambda{cost}\:+\lambda^{\mathrm{2}} }{dt}\:\:.{calculate}\:{I}_{{n}} \left(\lambda\right). \\ $$

Question Number 32340 Answers: 0 Comments: 0

find the value of ∫_0 ^∞ ((sin^3 t)/t^2 ) dt .

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{sin}^{\mathrm{3}} {t}}{{t}^{\mathrm{2}} }\:{dt}\:. \\ $$

Question Number 32339 Answers: 0 Comments: 1

calculate ∫_0 ^(+∞) ((th(3x) −th(2x))/x) dx .

$${calculate}\:\:\int_{\mathrm{0}} ^{+\infty} \:\:\:\frac{{th}\left(\mathrm{3}{x}\right)\:−{th}\left(\mathrm{2}{x}\right)}{{x}}\:{dx}\:. \\ $$

Question Number 32338 Answers: 0 Comments: 0

find the value of ∫_0 ^1 ((ln(t))/((1+t)(√(1−t^2 )))) dt.

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{ln}\left({t}\right)}{\left(\mathrm{1}+{t}\right)\sqrt{\mathrm{1}−{t}^{\mathrm{2}} }}\:{dt}. \\ $$

Question Number 32337 Answers: 0 Comments: 0

1)calculate ∫_a ^(+∞) (dx/((1+x^2 )(√(x^2 −a^2 )))) with a>0 2) find the value of ∫_2 ^(+∞) (dx/((1+x^2 )(√(x^2 −4)))) .

$$\left.\mathrm{1}\right){calculate}\:\int_{{a}} ^{+\infty} \:\:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\sqrt{{x}^{\mathrm{2}} \:−{a}^{\mathrm{2}} }}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\:\int_{\mathrm{2}} ^{+\infty} \:\:\:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\sqrt{{x}^{\mathrm{2}} \:−\mathrm{4}}}\:\:. \\ $$

Question Number 32335 Answers: 0 Comments: 0

let t≥0 and f(t) =(t/(√(1+t))) .prove that the sequence S_n = Σ_(k=1) ^n f((k/n^2 )) converges and find its limit.

$${let}\:{t}\geqslant\mathrm{0}\:{and}\:{f}\left({t}\right)\:=\frac{{t}}{\sqrt{\mathrm{1}+{t}}}\:.{prove}\:{that}\:{the}\:{sequence} \\ $$$${S}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{f}\left(\frac{{k}}{{n}^{\mathrm{2}} }\right)\:\:{converges}\:{and}\:{find}\:{its}\:{limit}. \\ $$

Question Number 32334 Answers: 0 Comments: 1

p is apolynomial having n roots (x_i ) with x_i ≠xj for i≠j calculate Σ_(k=1) ^n (1/(1−x_k )) .

$${p}\:{is}\:{apolynomial}\:{having}\:{n}\:{roots}\:\:\left({x}_{{i}} \right)\:{with}\:{x}_{{i}} \neq{xj}\:{for} \\ $$$${i}\neq{j}\:\:{calculate}\:\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\:\frac{\mathrm{1}}{\mathrm{1}−{x}_{{k}} }\:. \\ $$

Question Number 32333 Answers: 0 Comments: 1

decompose F(x) = (1/((1−x)^2 (1−x^2 ))) inside R(x).

$${decompose}\:{F}\left({x}\right)\:=\:\frac{\mathrm{1}}{\left(\mathrm{1}−{x}\right)^{\mathrm{2}} \left(\mathrm{1}−{x}^{\mathrm{2}} \right)}\:{inside}\:{R}\left({x}\right). \\ $$

Question Number 32332 Answers: 0 Comments: 0

calculate Σ_(p=1) ^n (p/(1+p^2 +p^4 )) .

$${calculate}\:\:\sum_{{p}=\mathrm{1}} ^{{n}} \:\:\:\frac{{p}}{\mathrm{1}+{p}^{\mathrm{2}} \:+{p}^{\mathrm{4}} }\:\:. \\ $$

Question Number 32331 Answers: 0 Comments: 0

p is a polynomial having n simples roots (x_i )_(1≤i≤n) prove that Σ_(k=1) ^n (1/(p^′ (x_k ))) =0

$${p}\:{is}\:{a}\:{polynomial}\:{having}\:{n}\:{simples}\:{roots}\:\left({x}_{{i}} \right)_{\mathrm{1}\leqslant{i}\leqslant{n}} \\ $$$${prove}\:{that}\:\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{{p}^{'} \left({x}_{{k}} \right)}\:=\mathrm{0} \\ $$

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