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Question Number 27379 Answers: 1 Comments: 1

let give f(x)= ∫_x ^(2x) (dt/(ln(1+t^2 ))) calculate f^′ (x).

$${let}\:{give}\:{f}\left({x}\right)=\:\:\int_{{x}} ^{\mathrm{2}{x}} \:\:\frac{{dt}}{{ln}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}\:\:{calculate}\:{f}^{'} \left({x}\right). \\ $$

Question Number 27376 Answers: 0 Comments: 2

Question Number 27365 Answers: 0 Comments: 0

Question Number 27345 Answers: 0 Comments: 1

prove that ∫_0 ^∞ (t^(x−1) /(e^t −1))dt =ξ(x)Γ(x) with ξ(x)= Σ_(n=1) ^∝ (1/n^x ) and Γ(x)=∫_0 ^∞ t^(x−1) e^(−t) dt ( x>1)

$${prove}\:{that}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}^{{x}−\mathrm{1}} }{{e}^{{t}} −\mathrm{1}}{dt}\:\:=\xi\left({x}\right)\Gamma\left({x}\right) \\ $$$${with}\:\xi\left({x}\right)=\:\sum_{{n}=\mathrm{1}} ^{\propto} \:\frac{\mathrm{1}}{{n}^{{x}} }\:\:\:{and}\:\Gamma\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} {dt} \\ $$$$\left(\:{x}>\mathrm{1}\right) \\ $$

Question Number 27343 Answers: 1 Comments: 7

let give A=(_(2 2) ^(1 2) ) find A^n and e^A and e^(tA) . we remind that e^A = Σ_ (A^n /(n!))

$${let}\:{give}\:{A}=\left(_{\mathrm{2}\:\:\:\:\:\:\:\mathrm{2}} ^{\mathrm{1}\:\:\:\:\:\:\mathrm{2}} \right)\:\:\:{find}\:\:{A}^{{n}} \:\:\:{and}\:\:{e}^{{A}} \\ $$$${and}\:\:{e}^{{tA}} \:\:\:\:\:\:.\:{we}\:{remind}\:{that}\:\:{e}^{{A}} =\:\sum_{} \:\frac{{A}^{{n}} }{{n}!} \\ $$

Question Number 27342 Answers: 0 Comments: 1

find f(x)= ∫_0 ^∞ (e^(−x(1+t^2 )) /(1+t^2 )) dt interms ofx with x≥0 and calculate ∫_0 ^∞ e^(−t^2 ) dt .

$${find}\:{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{x}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)} }{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt}\:\:{interms}\:{ofx} \\ $$$${with}\:{x}\geqslant\mathrm{0}\:\:\:{and}\:{calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{t}^{\mathrm{2}} } {dt}\:. \\ $$

Question Number 27341 Answers: 0 Comments: 0

prove that ∫_0 ^∞ e^(−(t^2 +(1/t^2 ))) dt is convergeny and find its value .

$${prove}\:{that}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−\left({t}^{\mathrm{2}} \:+\frac{\mathrm{1}}{{t}^{\mathrm{2}} }\right)} {dt}\:{is}\:{convergeny} \\ $$$${and}\:{find}\:{its}\:{value}\:. \\ $$

Question Number 27335 Answers: 0 Comments: 1

if 2 chords of ellipse have the same distance from the centre of ellipse and the eccentric angle of the end points of the chords are respectivly α β γ δ then prove that tan (α/2)×tan (β/2)×tan (γ/2)×tan (δ/2)=1

$${if}\:\mathrm{2}\:{chords}\:{of}\:{ellipse}\:{have}\:{the}\:{same} \\ $$$${distance}\:{from}\:{the}\:{centre}\:{of}\:{ellipse} \\ $$$${and}\:{the}\:{eccentric}\:{angle}\:{of}\:{the}\:{end}\:{points}\:{of}\:{the}\:{chords} \\ $$$${are}\:{respectivly}\:\alpha\:\beta\:\gamma\:\delta\:{then}\:{prove}\:{that} \\ $$$$\mathrm{tan}\:\frac{\alpha}{\mathrm{2}}×\mathrm{tan}\:\frac{\beta}{\mathrm{2}}×\mathrm{tan}\:\frac{\gamma}{\mathrm{2}}×\mathrm{tan}\:\frac{\delta}{\mathrm{2}}=\mathrm{1} \\ $$

Question Number 27334 Answers: 1 Comments: 0

(q_1 /q_2 )=((x/(0.8−x)))^2 ; x=?

$$\frac{\mathrm{q}_{\mathrm{1}} }{\mathrm{q}_{\mathrm{2}} }=\left(\frac{\mathrm{x}}{\mathrm{0}.\mathrm{8}−\mathrm{x}}\right)^{\mathrm{2}} \:\:\:\:;\:\boldsymbol{\mathrm{x}}=? \\ $$

Question Number 27332 Answers: 1 Comments: 1

Question Number 27328 Answers: 0 Comments: 1

Question Number 27327 Answers: 0 Comments: 4

lim_(n→∞) (x^(2n) /(1+∣x∣+x^(4n) ))

$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{x}^{\mathrm{2n}} }{\mathrm{1}+\mid\mathrm{x}\mid+\mathrm{x}^{\mathrm{4n}} } \\ $$

Question Number 27326 Answers: 1 Comments: 0

What is the relationship between the centre of gravity and the centre of mass?

$${What}\:{is}\:{the}\:{relationship}\:{between} \\ $$$${the}\:{centre}\:{of}\:{gravity}\:{and}\:{the}\:{centre}\:{of} \\ $$$${mass}? \\ $$

Question Number 27321 Answers: 1 Comments: 1

Question Number 27751 Answers: 0 Comments: 0

what is relation between in tensity of diffraction anx slit width

$${what}\:{is}\:{relation}\:{between}\:{in} \\ $$$${tensity}\:{of}\:{diffraction}\:{anx} \\ $$$${slit}\:{width} \\ $$

Question Number 27309 Answers: 1 Comments: 0

find the value of ∫_0 ^∝ (((−1)^([x]) )/((2x+1)^2 ))dx

$${find}\:{the}\:{value}\:\:{of}\:\:\int_{\mathrm{0}} ^{\propto} \:\:\:\frac{\left(−\mathrm{1}\right)^{\left[{x}\right]} }{\left(\mathrm{2}{x}+\mathrm{1}\right)^{\mathrm{2}} }{dx}\: \\ $$

Question Number 27303 Answers: 1 Comments: 0

Question Number 27300 Answers: 3 Comments: 1

Question Number 27296 Answers: 1 Comments: 1

((sin^2 3A)/(sin^2 A)) − ((cos^2 3A)/(cos^2 A)) =

$$\frac{\mathrm{sin}^{\mathrm{2}} \mathrm{3}{A}}{\mathrm{sin}^{\mathrm{2}} {A}}\:−\:\frac{\mathrm{cos}^{\mathrm{2}} \mathrm{3}{A}}{\mathrm{cos}^{\mathrm{2}} {A}}\:=\: \\ $$

Question Number 27295 Answers: 0 Comments: 1

The value of the integral ∫_( 0) ^π (1/(a^2 −2a cos x+1)) dx (a< 1) is

$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{integral} \\ $$$$\underset{\:\mathrm{0}} {\overset{\pi} {\int}}\:\:\frac{\mathrm{1}}{{a}^{\mathrm{2}} −\mathrm{2}{a}\:\mathrm{cos}\:{x}+\mathrm{1}}\:{dx}\:\:\left({a}<\:\mathrm{1}\right)\:\mathrm{is} \\ $$

Question Number 27294 Answers: 0 Comments: 1

∫_(1/e) ^(tan x) (t/(1+t^2 )) dt + ∫_(1/e) ^(cot x) (1/(t(1+t^2 ))) dt =

$$\underset{\mathrm{1}/{e}} {\overset{\mathrm{tan}\:{x}} {\int}}\frac{{t}}{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt}\:+\:\underset{\mathrm{1}/{e}} {\overset{\mathrm{cot}\:{x}} {\int}}\:\frac{\mathrm{1}}{{t}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}\:{dt}\:= \\ $$

Question Number 27293 Answers: 1 Comments: 0

L^(−1) ((s^3 /(s^4 +4)))=?

$${L}^{−\mathrm{1}} \left(\frac{{s}^{\mathrm{3}} }{{s}^{\mathrm{4}} +\mathrm{4}}\right)=? \\ $$

Question Number 27287 Answers: 0 Comments: 1

Question Number 27283 Answers: 0 Comments: 0

Question Number 27282 Answers: 0 Comments: 1

∫log(2+x^2 )dx

$$\int{log}\left(\mathrm{2}+{x}^{\mathrm{2}} \right){dx} \\ $$

Question Number 27281 Answers: 1 Comments: 1

∫_(−1) ^1 (x^2 +cos x) log (((2+x)/(2−x)))dx = 0

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

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