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Question Number 53464 Answers: 1 Comments: 1

let U_n = (((∫_0 ^n e^(−x^2 ) dx)^2 )/(∫_0 ^n e^(−nx^2 ) dx)) 1) calculate lim_(n→+∞) U_n 2) determne nature of Σ U_n and Σ U_n ^3 .

$${let}\:{U}_{{n}} =\:\frac{\left(\int_{\mathrm{0}} ^{{n}} \:{e}^{−{x}^{\mathrm{2}} } {dx}\right)^{\mathrm{2}} }{\int_{\mathrm{0}} ^{{n}} \:\:{e}^{−{nx}^{\mathrm{2}} } {dx}} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{lim}_{{n}\rightarrow+\infty} \:\:{U}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{determne}\:{nature}\:{of}\:\Sigma\:\:{U}_{{n}} \:\:{and}\:\Sigma\:{U}_{{n}} ^{\mathrm{3}} \:. \\ $$

Question Number 53463 Answers: 1 Comments: 1

1)let 0<θ<(π/2) and A(θ) =∫_0 ^(π/2) (dx/(√(x^2 +2sinθ x +1))) calculate A(θ) 2) calculate ∫_0 ^(π/2) (dx/(√(x^2 +(√2)x +1)))

$$\left.\mathrm{1}\right){let}\:\mathrm{0}<\theta<\frac{\pi}{\mathrm{2}}\:\:\:\:{and}\:\:{A}\left(\theta\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{dx}}{\sqrt{{x}^{\mathrm{2}} \:+\mathrm{2}{sin}\theta\:{x}\:+\mathrm{1}}} \\ $$$${calculate}\:{A}\left(\theta\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{dx}}{\sqrt{{x}^{\mathrm{2}} \:+\sqrt{\mathrm{2}}{x}\:+\mathrm{1}}} \\ $$

Question Number 53462 Answers: 1 Comments: 0

find ∫_(−(π/4)) ^(π/4) ((xsinx)/(cos^2 x))dx

$${find}\:\int_{−\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{xsinx}}{{cos}^{\mathrm{2}} {x}}{dx} \\ $$

Question Number 53455 Answers: 0 Comments: 1

Is it possible to calculate the derivative of a function involving complex numbers ? Thanks

$$\mathrm{Is}\:\mathrm{it}\:\mathrm{possible}\:\mathrm{to}\:\mathrm{calculate}\:\mathrm{the}\:\mathrm{derivative} \\ $$$$\mathrm{of}\:\mathrm{a}\:\mathrm{function}\:\mathrm{involving}\:\mathrm{complex} \\ $$$$\mathrm{numbers}\:?\:{Thanks} \\ $$

Question Number 53536 Answers: 1 Comments: 0

If [x] stands for the gratest integer function the value of ∫_4 ^(10) (([x^2 ])/([x^2 −28x+196]+[x^2 ])) dx is

$$\mathrm{If}\:\left[{x}\right]\:\mathrm{stands}\:\mathrm{for}\:\mathrm{the}\:\mathrm{gratest}\:\mathrm{integer}\:\mathrm{function} \\ $$$$\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\int_{\mathrm{4}} ^{\mathrm{10}} \frac{\left[{x}^{\mathrm{2}} \right]}{\left[{x}^{\mathrm{2}} −\mathrm{28}{x}+\mathrm{196}\right]+\left[{x}^{\mathrm{2}} \right]}\:{dx}\:\mathrm{is} \\ $$$$ \\ $$

Question Number 53450 Answers: 1 Comments: 1

Question Number 53447 Answers: 1 Comments: 0

Question Number 53426 Answers: 1 Comments: 1

The general solution of the equation (dy/dx)+ylnx=x^(−x) a)x^x (1−ce^x ) b)−x^(−x) (1+ce^(2x) ) c)−x^(−x) (1−ce^x )

$${The}\:{general}\:{solution}\:{of}\:{the}\:{equation} \\ $$$$\frac{{dy}}{{dx}}+{ylnx}={x}^{−{x}} \\ $$$$\left.{a}\left.\right){x}^{{x}} \left(\mathrm{1}−{ce}^{{x}} \right)\:\:{b}\right)−{x}^{−{x}} \left(\mathrm{1}+{ce}^{\mathrm{2}{x}} \right) \\ $$$$\left.{c}\right)−{x}^{−{x}} \left(\mathrm{1}−{ce}^{{x}} \right) \\ $$

Question Number 53422 Answers: 4 Comments: 2

Question Number 53418 Answers: 0 Comments: 1

find ∫_0 ^π (x/(2+cosx sinx))dx

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

Question Number 53391 Answers: 1 Comments: 1

if A+B+C=2S, prove that 4 sin S sin(S−A)sin(S−B)sin(S−C) =1−cos^2 A−cos^2 B−cos^2 C+2 cos A cos B cos C

$${if}\:{A}+{B}+{C}=\mathrm{2}{S},\:{prove}\:{that}\: \\ $$$$\mathrm{4}\:\mathrm{sin}\:{S}\:\mathrm{sin}\left({S}−{A}\right)\mathrm{sin}\left({S}−{B}\right)\mathrm{sin}\left({S}−{C}\right) \\ $$$$=\mathrm{1}−\mathrm{cos}^{\mathrm{2}} \:{A}−\mathrm{cos}^{\mathrm{2}} \:{B}−\mathrm{cos}^{\mathrm{2}} \:{C}+\mathrm{2}\:\mathrm{cos}\:{A}\:\mathrm{cos}\:{B}\:\mathrm{cos}\:{C}\: \\ $$

Question Number 53386 Answers: 0 Comments: 6

∫_( 0) ^1 (dx/(e^x + e^(−x) )) = tan^(−1) e− (π/4)

$$\underset{\:\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\:\:\frac{{dx}}{{e}^{{x}} +\:{e}^{−{x}} }\:=\:\mathrm{tan}^{−\mathrm{1}} {e}−\:\frac{\pi}{\mathrm{4}} \\ $$

Question Number 53385 Answers: 1 Comments: 0

The value of the integral ∫_( 0) ^π ((x dx)/(1+cos α sin α)) , 0< α<π is

$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{integral}\:\underset{\:\mathrm{0}} {\overset{\pi} {\int}}\:\frac{{x}\:{dx}}{\mathrm{1}+\mathrm{cos}\:\alpha\:\mathrm{sin}\:\alpha}\:, \\ $$$$\mathrm{0}<\:\alpha<\pi\:\:\:\mathrm{is} \\ $$

Question Number 53384 Answers: 1 Comments: 1

∫_( 0) ^1 (1+e^(−x^2 ) ) dx =

$$\:\underset{\:\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\left(\mathrm{1}+{e}^{−{x}^{\mathrm{2}} } \right)\:{dx}\:= \\ $$

Question Number 53383 Answers: 1 Comments: 1

If ∫(1/((sin x+4)(sin x−1)))dx = A(1/(tan (x/2)−1))+B tan^(−1) (f(x))+C, then

$$\mathrm{If}\:\int\frac{\mathrm{1}}{\left(\mathrm{sin}\:{x}+\mathrm{4}\right)\left(\mathrm{sin}\:{x}−\mathrm{1}\right)}{dx} \\ $$$$\:\:\:\:\:\:\:\:\:=\:{A}\frac{\mathrm{1}}{\mathrm{tan}\:\frac{{x}}{\mathrm{2}}−\mathrm{1}}+{B}\:\mathrm{tan}^{−\mathrm{1}} \left({f}\left({x}\right)\right)+{C},\:\mathrm{then} \\ $$

Question Number 53382 Answers: 1 Comments: 0

∫ cos^3 x e^(log (sin x)) dx =

$$\int\:\mathrm{cos}^{\mathrm{3}} {x}\:{e}^{\mathrm{log}\:\left(\mathrm{sin}\:{x}\right)} {dx}\:= \\ $$

Question Number 53381 Answers: 1 Comments: 0

∫ (1/([(x−1)^3 (x+2)^5 ]^(1/4) )) dx =

$$\int\:\frac{\mathrm{1}}{\left[\left({x}−\mathrm{1}\right)^{\mathrm{3}} \left({x}+\mathrm{2}\right)^{\mathrm{5}} \right]^{\mathrm{1}/\mathrm{4}} }\:{dx}\:= \\ $$

Question Number 53378 Answers: 1 Comments: 0

if u=e^(xyz) then u_(xyx) =? a)u((xyz)^2 +3xyz+1) b)u(3(xyz)^2 +1) c)u((xyz)^2 +2yz+1) please help

$${if}\:{u}={e}^{{xyz}} \:{then}\:{u}_{{xyx}} =? \\ $$$$\left.{a}\left.\right){u}\left(\left({xyz}\right)^{\mathrm{2}} +\mathrm{3}{xyz}+\mathrm{1}\right)\:{b}\right){u}\left(\mathrm{3}\left({xyz}\right)^{\mathrm{2}} +\mathrm{1}\right) \\ $$$$\left.{c}\right){u}\left(\left({xyz}\right)^{\mathrm{2}} +\mathrm{2}{yz}+\mathrm{1}\right) \\ $$$$ \\ $$$${please}\:{help} \\ $$

Question Number 53376 Answers: 2 Comments: 1

Question Number 53359 Answers: 2 Comments: 0

∫ ((x^3 −1)/(x^3 +x)) dx =

$$\int\:\:\frac{{x}^{\mathrm{3}} −\mathrm{1}}{{x}^{\mathrm{3}} +{x}}\:{dx}\:= \\ $$

Question Number 53358 Answers: 1 Comments: 0

∫ (1/(√(sin^3 x cos x))) dx =

$$\int\:\:\frac{\mathrm{1}}{\sqrt{\mathrm{sin}^{\mathrm{3}} {x}\:\mathrm{cos}\:{x}}}\:{dx}\:= \\ $$

Question Number 53357 Answers: 1 Comments: 0

∫ (((x^4 −x)^(1/4) )/x^5 ) dx =

$$\int\:\:\frac{\left({x}^{\mathrm{4}} −{x}\right)^{\mathrm{1}/\mathrm{4}} }{{x}^{\mathrm{5}} }\:{dx}\:= \\ $$

Question Number 53353 Answers: 1 Comments: 0

Question Number 53351 Answers: 0 Comments: 1

Question Number 53349 Answers: 0 Comments: 2

Question Number 53342 Answers: 0 Comments: 4

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