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Question Number 53492 Answers: 1 Comments: 1

Question Number 53491 Answers: 3 Comments: 0

(((1+x)/(√x)))^2 +2a(((1+x)/(√x)))+1=0 solve for x.

$$\left(\frac{\mathrm{1}+{x}}{\sqrt{{x}}}\right)^{\mathrm{2}} +\mathrm{2}{a}\left(\frac{\mathrm{1}+{x}}{\sqrt{{x}}}\right)+\mathrm{1}=\mathrm{0} \\ $$$${solve}\:{for}\:{x}. \\ $$

Question Number 53483 Answers: 0 Comments: 9

Question Number 53477 Answers: 1 Comments: 1

let f(a)=∫_0 ^1 (dt/((√(x+a)) +3)) 1) calculate f(a) 2) find also ∫_0 ^1 (dt/((√(x+a))((√(x+a)) +3)^2 )) 3) find the values of integrals ∫_0 ^1 (dt/((√(x+1))+3)) and ∫_0 ^1 (dt/((√(x+1))((√(x+1))+3)^2 ))

$${let}\:{f}\left({a}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dt}}{\sqrt{{x}+{a}}\:+\mathrm{3}} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{also}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dt}}{\sqrt{{x}+{a}}\left(\sqrt{{x}+{a}}\:+\mathrm{3}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{values}\:{of}\:{integrals}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dt}}{\sqrt{{x}+\mathrm{1}}+\mathrm{3}}\:\:{and}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dt}}{\sqrt{{x}+\mathrm{1}}\left(\sqrt{{x}+\mathrm{1}}+\mathrm{3}\right)^{\mathrm{2}} } \\ $$

Question Number 53476 Answers: 0 Comments: 0

let f(x)=∫_0 ^x t(√(2t−1))dt calculate ∣sup_(1≤x≤2) f(x) −inf_(1≤x≤2) f(x)∣

$${let}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{{x}} \:{t}\sqrt{\mathrm{2}{t}−\mathrm{1}}{dt}\:\:\:\:{calculate}\:\mid{sup}_{\mathrm{1}\leqslant{x}\leqslant\mathrm{2}} \:{f}\left({x}\right)\:−{inf}_{\mathrm{1}\leqslant{x}\leqslant\mathrm{2}} {f}\left({x}\right)\mid \\ $$

Question Number 53474 Answers: 1 Comments: 0

calculate ∫_0 ^1 ((5^(2x+1) −2^(2x−1) )/(10^x )) dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\mathrm{5}^{\mathrm{2}{x}+\mathrm{1}} \:−\mathrm{2}^{\mathrm{2}{x}−\mathrm{1}} }{\mathrm{10}^{{x}} }\:{dx} \\ $$

Question Number 53472 Answers: 1 Comments: 2

Question Number 53471 Answers: 1 Comments: 3

1)find U_n = ∫_0 ^(π/4) tan^n tdt with n integr . 2) find lim_(n→+∞) U_n 3) calculate Σ_(n=0) ^∞ U_n

$$\left.\mathrm{1}\right){find}\:\:{U}_{{n}} =\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:{tan}^{{n}} {tdt}\:\:\:{with}\:{n}\:{integr}\:. \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} {U}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:{U}_{{n}} \\ $$$$ \\ $$

Question Number 53470 Answers: 0 Comments: 1

find Vn=∫_(1/n) ^((an−1)/n) ((√x)/(√(a−(√x)+x)))dx

$${find}\:\:{Vn}=\int_{\frac{\mathrm{1}}{{n}}} ^{\frac{{an}−\mathrm{1}}{{n}}} \:\frac{\sqrt{{x}}}{\sqrt{{a}−\sqrt{{x}}+{x}}}{dx} \\ $$$$ \\ $$

Question Number 53468 Answers: 1 Comments: 3

Question Number 53467 Answers: 1 Comments: 0

let A_(n m) =∫_0 ^1 x^n (1−x)^m dx with n and n integrs naturals 1) calculate A_(n m) by using factoriels 2) find Σ_(n,m) A_(nm)

$${let}\:{A}_{{n}\:{m}} \:\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{{n}} \left(\mathrm{1}−{x}\right)^{{m}} {dx}\:\:{with}\:{n}\:{and}\:{n}\:{integrs}\:{naturals} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}_{{n}\:{m}} \:\:{by}\:{using}\:{factoriels} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\sum_{{n},{m}} \:{A}_{{nm}} \\ $$

Question Number 53466 Answers: 1 Comments: 0

let U_n =(1/n){Π_(k=1) ^n (n+k)}^(1/n) find lim_(n→+∞) U_n

$${let}\:{U}_{{n}} =\frac{\mathrm{1}}{{n}}\left\{\prod_{{k}=\mathrm{1}} ^{{n}} \left({n}+{k}\right)\right\}^{\frac{\mathrm{1}}{{n}}} \\ $$$${find}\:{lim}_{{n}\rightarrow+\infty} \:{U}_{{n}} \\ $$

Question Number 53465 Answers: 1 Comments: 1

find ∫_(−(π/2)) ^(π/2) (√(cosx −cos^3 x))dx

$${find}\:\int_{−\frac{\pi}{\mathrm{2}}} ^{\frac{\pi}{\mathrm{2}}} \sqrt{{cosx}\:−{cos}^{\mathrm{3}} {x}}{dx} \\ $$

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

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

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