Question and Answers Forum

All Questions Topic List

AllQuestion and Answers: Page 1453

Question Number 63720 Answers: 1 Comments: 2

calculate ∫(√((x−3)(2−x)))dx

$${calculate}\:\int\sqrt{\left({x}−\mathrm{3}\right)\left(\mathrm{2}−{x}\right)}{dx} \\ $$

Question Number 63711 Answers: 1 Comments: 1

calculate ∫_0 ^π (dx/((√3)cosx +(√2)sinx))

$${calculate}\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{dx}}{\sqrt{\mathrm{3}}{cosx}\:+\sqrt{\mathrm{2}}{sinx}} \\ $$

Question Number 63710 Answers: 0 Comments: 1

Question Number 63703 Answers: 1 Comments: 2

sin^3 x+cos^3 x=1−(1/2)sin2x :x∈[0,2π]. find x

$${sin}^{\mathrm{3}} {x}+{cos}^{\mathrm{3}} {x}=\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}{sin}\mathrm{2}{x}\:\::{x}\in\left[\mathrm{0},\mathrm{2}\pi\right]. \\ $$$${find}\:\:{x} \\ $$

Question Number 63700 Answers: 0 Comments: 0

Question Number 63698 Answers: 0 Comments: 0

Question Number 63693 Answers: 1 Comments: 2

find the general solution for sin5θ+sin3θ= 1

$${find}\:{the}\:{general}\:{solution}\:{for}\: \\ $$$$\:{sin}\mathrm{5}\theta+{sin}\mathrm{3}\theta=\:\mathrm{1} \\ $$

Question Number 63689 Answers: 0 Comments: 3

Show that if a∣b then an∣bn

$${Show}\:{that}\:\:{if}\:\:{a}\mid{b}\:\:{then}\:{an}\mid{bn} \\ $$

Question Number 63684 Answers: 0 Comments: 0

cot 118

$$\mathrm{cot}\:\mathrm{118} \\ $$

Question Number 63682 Answers: 0 Comments: 0

given diameter 25mm half of the drill point angle =60 cutting velocity=44000mm/minute length=60mm feedrate=0.25mm/revolution determine the time needed to drill a through hole

$${given}\:{diameter}\:\mathrm{25}{mm} \\ $$$${half}\:{of}\:{the}\:{drill}\:{point}\:{angle}\:=\mathrm{60} \\ $$$${cutting}\:{velocity}=\mathrm{44000}{mm}/{minute} \\ $$$${length}=\mathrm{60}{mm} \\ $$$${feedrate}=\mathrm{0}.\mathrm{25}{mm}/{revolution} \\ $$$${determine}\:{the}\:{time}\:{needed}\:{to}\:{drill}\:{a}\:{through}\:{hole} \\ $$

Question Number 63678 Answers: 0 Comments: 1

Question Number 63674 Answers: 0 Comments: 2

Show that the number 122^n − 102^n − 21^n is always one less than a multiple of 2020. For every positive integer n.

$$\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{number}\:\:\mathrm{122}^{\mathrm{n}} \:−\:\mathrm{102}^{\mathrm{n}} \:−\:\mathrm{21}^{\mathrm{n}} \:\:\mathrm{is}\:\mathrm{always}\:\mathrm{one}\:\mathrm{less}\:\mathrm{than}\:\mathrm{a} \\ $$$$\:\mathrm{multiple}\:\mathrm{of}\:\:\mathrm{2020}.\:\:\mathrm{For}\:\mathrm{every}\:\mathrm{positive}\:\mathrm{integer}\:\:\mathrm{n}. \\ $$

Question Number 63667 Answers: 0 Comments: 3

1) calculate ∫_0 ^(2π) (dt/(cost +x sint)) wih x from R. 2) calculate ∫_0 ^(2π) ((sint)/((cost +xsint)^2 ))dt 3) find[the value of ∫_0 ^(2π) (dt/(cos(2t)+2sin(2t)))

$$\left.\mathrm{1}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{dt}}{{cost}\:+{x}\:{sint}}\:\:\:{wih}\:{x}\:{from}\:{R}. \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{sint}}{\left({cost}\:+{xsint}\right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{3}\right)\:{find}\left[{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{dt}}{{cos}\left(\mathrm{2}{t}\right)+\mathrm{2}{sin}\left(\mathrm{2}{t}\right)}\right. \\ $$

Question Number 63666 Answers: 0 Comments: 3

calculate ∫_0 ^(2π) (dx/(2sinx +cosx))

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\frac{{dx}}{\mathrm{2}{sinx}\:+{cosx}} \\ $$

Question Number 63665 Answers: 0 Comments: 1

find the value of Σ_(n=1) ^∞ (((−1)^n )/(n^2 (n+1)^3 ))

$${find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)^{\mathrm{3}} } \\ $$

Question Number 63664 Answers: 0 Comments: 6

let f(x)=∫_0 ^∞ (t^(a−1) /(x+t^n )) dt with 0<a<1 and x>0 and n≥2 1) determine a explicit form of f(x) 2) calculate g(x) =∫_0 ^∞ (t^(a−1) /((x+t^n )^2 )) dt 3) find f^((k)) (x) at form of integrals 4) calculate ∫_0 ^∞ (t^(a−1) /(9+t^2 )) dt and ∫_0 ^∞ (t^(a−1) /((9+t^2 )^2 )) 5) calculate U_n =∫_0 ^∞ (t^((1/n)−1) /(2^n +t^n )) dt and study the convergence of Σ U_n

$${let}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}^{{a}−\mathrm{1}} }{{x}+{t}^{{n}} }\:{dt}\:\:\:{with}\:\mathrm{0}<{a}<\mathrm{1}\:\:{and}\:\:{x}>\mathrm{0}\:{and}\:{n}\geqslant\mathrm{2} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}^{{a}−\mathrm{1}} }{\left({x}+{t}^{{n}} \right)^{\mathrm{2}} }\:{dt} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{f}^{\left({k}\right)} \left({x}\right)\:{at}\:{form}\:{of}\:{integrals} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}^{{a}−\mathrm{1}} }{\mathrm{9}+{t}^{\mathrm{2}} }\:{dt}\:\:\:\:{and}\:\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{t}^{{a}−\mathrm{1}} }{\left(\mathrm{9}+{t}^{\mathrm{2}} \right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{5}\right)\:{calculate}\:\:{U}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{t}^{\frac{\mathrm{1}}{{n}}−\mathrm{1}} }{\mathrm{2}^{{n}} \:+{t}^{{n}} }\:{dt}\:\:{and}\:{study}\:{the}\:{convergence}\:{of}\:\Sigma\:{U}_{{n}} \\ $$

Question Number 63663 Answers: 1 Comments: 0

Question Number 63662 Answers: 0 Comments: 1

let A_n =∫_0 ^∞ (x^(a−1) /(1+x^n ))dx with n integr and n≥2 and 0<a<1 1) calculate A_n 2) find the values of ∫_0 ^∞ (x^(a−1) /(1+x^2 ))dx and ∫_0 ^∞ (x^(a−1) /(1+x^3 ))dx 3)calculate ∫_0 ^∞ (dx/((√x)(1+x^4 ))) and ∫_0 ^∞ (dx/((^3 (√x^2 ))(1+x^4 )))

$$\:{let}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{x}^{{a}−\mathrm{1}} }{\mathrm{1}+{x}^{{n}} }{dx}\:\:{with}\:{n}\:{integr}\:{and}\:{n}\geqslant\mathrm{2}\:\:{and}\:\mathrm{0}<{a}<\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}^{{a}−\mathrm{1}} }{\mathrm{1}+{x}^{\mathrm{2}} }{dx}\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}^{{a}−\mathrm{1}} }{\mathrm{1}+{x}^{\mathrm{3}} }{dx} \\ $$$$\left.\mathrm{3}\right){calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{dx}}{\sqrt{{x}}\left(\mathrm{1}+{x}^{\mathrm{4}} \right)}\:\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left(^{\mathrm{3}} \sqrt{{x}^{\mathrm{2}} }\right)\left(\mathrm{1}+{x}^{\mathrm{4}} \right)} \\ $$

Question Number 63661 Answers: 0 Comments: 1

let 0<a<1 find the valueof ∫_0 ^∞ (t^(a−1) /(1+t^2 ))dt

$${let}\:\mathrm{0}<{a}<\mathrm{1}\:{find}\:{the}\:{valueof}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}^{{a}−\mathrm{1}} }{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$

Question Number 63642 Answers: 2 Comments: 1