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Question Number 43147    Answers: 0   Comments: 2

If ∫_0 ^∞ e^(−x^2 ) dx = ((√π)/(2 )) , then prove that ∫_0 ^∞ e^(−ax^2 ) dx = (√(π/(4a))) where a>0.

$$\mathrm{If}\:\:\:\int_{\mathrm{0}} ^{\infty} \:\mathrm{e}^{−{x}^{\mathrm{2}} } {dx}\:=\:\frac{\sqrt{\pi}}{\mathrm{2}\:}\:, \\ $$$$\mathrm{then}\:{prove}\:{that}\:\int_{\mathrm{0}} ^{\infty} \mathrm{e}^{−\mathrm{a}{x}^{\mathrm{2}} } {dx}\:=\:\sqrt{\frac{\pi}{\mathrm{4a}}} \\ $$$$\mathrm{where}\:\mathrm{a}>\mathrm{0}. \\ $$

Question Number 43145    Answers: 1   Comments: 4

∫_0 ^∞ [ 2e^(−x) ]dx = ? where [.]= gif.

$$\int_{\mathrm{0}} ^{\infty} \:\left[\:\mathrm{2e}^{−{x}} \right]{dx}\:=\:?\: \\ $$$${where}\:\left[.\right]=\:{gif}. \\ $$

Question Number 43125    Answers: 1   Comments: 1

Question Number 43100    Answers: 1   Comments: 2

let f(x) = ∫_0 ^(π/2) ((cosθ)/(1+xsinθ))dθ 1) determine a explicit form of f(x) 2) calculate ∫_0 ^(π/2) ((sin(2θ))/((1+xsinθ)^2 ))dθ 3) find the values of ∫_0 ^(π/2) ((cosθ)/(1+2cosθ))dθ and ∫_0 ^(π/2) ((sin(2θ))/((1+3sinθ)^2 ))dθ .

$${let}\:{f}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\:\frac{{cos}\theta}{\mathrm{1}+{xsin}\theta}{d}\theta \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{sin}\left(\mathrm{2}\theta\right)}{\left(\mathrm{1}+{xsin}\theta\right)^{\mathrm{2}} }{d}\theta \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{values}\:{of}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{cos}\theta}{\mathrm{1}+\mathrm{2}{cos}\theta}{d}\theta\:\:\:{and}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\:\frac{{sin}\left(\mathrm{2}\theta\right)}{\left(\mathrm{1}+\mathrm{3}{sin}\theta\right)^{\mathrm{2}} }{d}\theta\:. \\ $$

Question Number 43064    Answers: 0   Comments: 8

Question Number 43058    Answers: 0   Comments: 1

calculate ∫_0 ^∞ ((x sin(((πx)/2)))/({1+(x+1)^2 }{1+(x−1)^2 }))dx

$${calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{x}\:{sin}\left(\frac{\pi{x}}{\mathrm{2}}\right)}{\left\{\mathrm{1}+\left({x}+\mathrm{1}\right)^{\mathrm{2}} \right\}\left\{\mathrm{1}+\left({x}−\mathrm{1}\right)^{\mathrm{2}} \right\}}{dx} \\ $$

Question Number 43057    Answers: 0   Comments: 1

find the value of I = ∫_0 ^∞ ((cos(πx^2 ) −sin(πx^2 ))/((1+x^2 )^2 ))dx

$${find}\:{the}\:{value}\:{of}\:\:\:{I}\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{cos}\left(\pi{x}^{\mathrm{2}} \right)\:−{sin}\left(\pi{x}^{\mathrm{2}} \right)}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx} \\ $$

Question Number 43027    Answers: 2   Comments: 1

Question Number 43008    Answers: 1   Comments: 0

(y′)^2 =−1+sin x y=?

$$\left({y}'\right)^{\mathrm{2}} =−\mathrm{1}+\mathrm{sin}\:{x} \\ $$$${y}=? \\ $$

Question Number 42994    Answers: 0   Comments: 5

∫(√(1+((cos x)/(4tan x))))dx=?

$$\int\sqrt{\mathrm{1}+\frac{\mathrm{cos}\:{x}}{\mathrm{4tan}\:{x}}}{dx}=? \\ $$

Question Number 42945    Answers: 2   Comments: 12

∫_0 ^( π/2) (dx/(√(sin x))) = ?

$$\int_{\mathrm{0}} ^{\:\:\pi/\mathrm{2}} \frac{{dx}}{\sqrt{\mathrm{sin}\:{x}}}\:=\:? \\ $$

Question Number 42870    Answers: 0   Comments: 0

let 0<x<1 and Γ(x) =∫_0 ^∞ t^(x−1) e^(−t) dt 1) prove that Γ(x).Γ(1−x) =(π/(sin(πx))) (compliments formulae) 2) calculate Γ(n) and Γ(n+(1/2)) with n from N.

$${let}\:\mathrm{0}<{x}<\mathrm{1}\:\:{and}\:\Gamma\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} \:{dt}\: \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\Gamma\left({x}\right).\Gamma\left(\mathrm{1}−{x}\right)\:=\frac{\pi}{{sin}\left(\pi{x}\right)}\:\:\:\left({compliments}\:{formulae}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\Gamma\left({n}\right)\:{and}\:\Gamma\left({n}+\frac{\mathrm{1}}{\mathrm{2}}\right)\:{with}\:{n}\:{from}\:{N}. \\ $$

Question Number 42823    Answers: 1   Comments: 1

Evaluate : ∫_(−5) ^( 5) x^2 [x+(1/2)]dx = ? where [.]= greatest integer function

$$\mathrm{Evaluate}\:: \\ $$$$\int_{−\mathrm{5}} ^{\:\mathrm{5}} \:{x}^{\mathrm{2}} \left[{x}+\frac{\mathrm{1}}{\mathrm{2}}\right]{dx}\:=\:\:? \\ $$$${where}\:\left[.\right]=\:{greatest}\:{integer}\:{function} \\ $$

Question Number 42812    Answers: 0   Comments: 1

study the convervence of ∫_1 ^(+∞) ((arctan(x−1))/((x^2 −1)^(4/3) )) dx

$${study}\:{the}\:{convervence}\:{of}\:\:\int_{\mathrm{1}} ^{+\infty} \:\:\frac{{arctan}\left({x}−\mathrm{1}\right)}{\left({x}^{\mathrm{2}} −\mathrm{1}\right)^{\frac{\mathrm{4}}{\mathrm{3}}} }\:{dx} \\ $$

Question Number 42810    Answers: 1   Comments: 1

find ∫_0 ^∞ (x^5 /(1+x^7 ))dx .

$${find}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{x}^{\mathrm{5}} }{\mathrm{1}+{x}^{\mathrm{7}} }{dx}\:\:. \\ $$

Question Number 42809    Answers: 1   Comments: 1

calculate ∫_0 ^∞ ((tdt)/((1+t^4 )^2 ))

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{tdt}}{\left(\mathrm{1}+{t}^{\mathrm{4}} \right)^{\mathrm{2}} } \\ $$

Question Number 42806    Answers: 0   Comments: 0

let u_n = ∫_n ^(n+2) (((t+n)^(1/4) )/t^(1/3) )dt find lim_(n→+∞) u_n

$${let}\:\:{u}_{{n}} =\:\int_{{n}} ^{{n}+\mathrm{2}} \:\:\:\frac{\left({t}+{n}\right)^{\frac{\mathrm{1}}{\mathrm{4}}} }{{t}^{\frac{\mathrm{1}}{\mathrm{3}}} }{dt} \\ $$$${find}\:{lim}_{{n}\rightarrow+\infty} \:{u}_{{n}} \\ $$

Question Number 42804    Answers: 1   Comments: 1

calculate ∫_(1/2) ^1 (dx/((√(4x^2 −1)) +(√(4x^2 +1))))

$${calculate}\:\:\int_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\mathrm{1}} \:\:\:\:\:\frac{{dx}}{\sqrt{\mathrm{4}{x}^{\mathrm{2}} \:−\mathrm{1}}\:+\sqrt{\mathrm{4}{x}^{\mathrm{2}} \:+\mathrm{1}}} \\ $$

Question Number 42803    Answers: 1   Comments: 0

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

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

Question Number 42802    Answers: 0   Comments: 1

calculate ∫_(1/2) ^(5/4) (x^3 /(√(2+x−x^2 )))dx

$${calculate}\:\:\int_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\frac{\mathrm{5}}{\mathrm{4}}} \:\:\:\frac{{x}^{\mathrm{3}} }{\sqrt{\mathrm{2}+{x}−{x}^{\mathrm{2}} }}{dx} \\ $$

Question Number 42801    Answers: 1   Comments: 1

find f(x) = ∫_(π/4) ^(π/3) ((cosxdx)/(2cos^2 x +sin^2 x +1))

$${find}\:{f}\left({x}\right)\:=\:\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \:\:\:\:\frac{{cosxdx}}{\mathrm{2}{cos}^{\mathrm{2}} {x}\:+{sin}^{\mathrm{2}} {x}\:+\mathrm{1}} \\ $$

Question Number 42800    Answers: 1   Comments: 0

find ∫ ((sinx)/(1+2 cosx))dx

$${find}\:\int\:\:\:\:\:\frac{{sinx}}{\mathrm{1}+\mathrm{2}\:{cosx}}{dx} \\ $$

Question Number 42799    Answers: 0   Comments: 2

let I = ∫_0 ^(π/8) e^(−2t) cos^4 t and J =∫_0 ^(π/8) e^(−2t) sin^4 dt find the values of I andJ .

$${let}\:{I}\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{8}}} \:\:{e}^{−\mathrm{2}{t}} \:{cos}^{\mathrm{4}} {t}\:\:\:\:{and}\:{J}\:\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{8}}} \:{e}^{−\mathrm{2}{t}} \:{sin}^{\mathrm{4}} {dt} \\ $$$${find}\:{the}\:{values}\:{of}\:{I}\:{andJ}\:. \\ $$

Question Number 42798    Answers: 1   Comments: 2

find I_n = ∫_0 ^1 x^n (√(1−x^2 ))dx

$${find}\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{{n}} \sqrt{\mathrm{1}−{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 42797    Answers: 0   Comments: 1

let u_k = ∫_(−(π/2) +kπ) ^(−(π/2) +(k+1)π) e^(−t) cost dt 1) calculate u_k 2) let A_n =Σ_(k=0) ^n u_k find lim_(n→+∞) A_n

$${let}\:{u}_{{k}} =\:\int_{−\frac{\pi}{\mathrm{2}}\:+{k}\pi} ^{−\frac{\pi}{\mathrm{2}}\:+\left({k}+\mathrm{1}\right)\pi} \:\:{e}^{−{t}} \:{cost}\:{dt} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{u}_{{k}} \\ $$$$\left.\mathrm{2}\right)\:{let}\:{A}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:{u}_{{k}} \:\:\:\:\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} \\ $$

Question Number 42796    Answers: 1   Comments: 1

calculate I = ∫_0 ^1 (x^2 /(1+x^2 )) arctan(x)dx

$${calculate}\:{I}\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{x}^{\mathrm{2}} }{\mathrm{1}+{x}^{\mathrm{2}} }\:{arctan}\left({x}\right){dx} \\ $$

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