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Question Number 84680 Answers: 1 Comments: 0

show that ∫_0 ^1 ∫_0 ^1 ∫_0 ^1 ((log(xyz))/((1+x^2 )(1+y^2 )(1+z^2 ))) dx dy dz=((−3π^2 G)/(16))

$${show}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} \frac{{log}\left({xyz}\right)}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{y}^{\mathrm{2}} \right)\left(\mathrm{1}+{z}^{\mathrm{2}} \right)}\:{dx}\:{dy}\:{dz}=\frac{−\mathrm{3}\pi^{\mathrm{2}} {G}}{\mathrm{16}} \\ $$

Question Number 84677 Answers: 0 Comments: 1

Σ_(k = 1) ^∞ (k^2 /2^k ) ?

$$\underset{\mathrm{k}\:=\:\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{k}^{\mathrm{2}} }{\mathrm{2}^{\mathrm{k}} }\:?\: \\ $$

Question Number 84667 Answers: 0 Comments: 1

Question Number 84666 Answers: 1 Comments: 2

∫ x sin^(−1) (x) dx

$$\int\:{x}\:\mathrm{sin}^{−\mathrm{1}} \left({x}\right)\:{dx}\: \\ $$

Question Number 84674 Answers: 0 Comments: 1

((6−log_(16) (x^4 ))/(3+2log_(16) (x^2 ))) < 2

$$\frac{\mathrm{6}−\mathrm{log}_{\mathrm{16}} \:\left(\mathrm{x}^{\mathrm{4}} \right)}{\mathrm{3}+\mathrm{2log}_{\mathrm{16}} \left(\mathrm{x}^{\mathrm{2}} \right)}\:<\:\mathrm{2} \\ $$

Question Number 84655 Answers: 2 Comments: 1

Question Number 84651 Answers: 1 Comments: 0

A person stands in the diagonal produced of the square base of a church tower, at a distance 2a from it, and observes the angle of elevation of each of the two outer corners of the top of the tower to be 30°, while that of the nearest corner is 45°. Find the breadth of the tower.

$${A}\:{person}\:{stands}\:{in}\:{the}\:{diagonal} \\ $$$${produced}\:{of}\:{the}\:{square}\:{base}\:{of}\:{a} \\ $$$${church}\:{tower},\:{at}\:{a}\:{distance}\:\mathrm{2}{a} \\ $$$${from}\:{it},\:{and}\:{observes}\:{the}\:{angle} \\ $$$${of}\:{elevation}\:{of}\:{each}\:{of}\:{the}\:{two} \\ $$$${outer}\:{corners}\:{of}\:{the}\:{top}\:{of}\:{the} \\ $$$${tower}\:{to}\:{be}\:\mathrm{30}°,\:{while}\:{that}\:{of}\:{the} \\ $$$${nearest}\:{corner}\:{is}\:\mathrm{45}°.\:{Find}\:{the} \\ $$$${breadth}\:{of}\:{the}\:{tower}. \\ $$

Question Number 84641 Answers: 1 Comments: 1

Question Number 84637 Answers: 0 Comments: 5

prove that lim_(x→∞) (1 + (1/x))^x =e

$$\mathrm{prove}\:\mathrm{that}\:\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(\mathrm{1}\:+\:\frac{\mathrm{1}}{{x}}\right)^{{x}} \:={e} \\ $$

Question Number 84632 Answers: 0 Comments: 0

Question Number 84628 Answers: 1 Comments: 1

Question Number 84627 Answers: 1 Comments: 2

1)∣sec(x)∣<2tan(x) on[0,2π] 2)find the cirtical points and the range of f(x)=∣x−3∣+∣2x+1∣

$$\left.\mathrm{1}\right)\mid{sec}\left({x}\right)\mid<\mathrm{2}{tan}\left({x}\right)\:{on}\left[\mathrm{0},\mathrm{2}\pi\right] \\ $$$$ \\ $$$$\left.\mathrm{2}\right){find}\:{the}\:{cirtical}\:{points}\:{and}\:{the}\:{range}\:{of} \\ $$$${f}\left({x}\right)=\mid{x}−\mathrm{3}\mid+\mid\mathrm{2}{x}+\mathrm{1}\mid \\ $$$$ \\ $$

Question Number 84624 Answers: 1 Comments: 0

find grad r^m where r=x^2 +y^2 +z^2

$$\mathrm{find}\:\mathrm{grad}\:\mathrm{r}^{\mathrm{m}} \:\:\mathrm{where}\:\mathrm{r}=\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{z}^{\mathrm{2}} \\ $$

Question Number 84607 Answers: 3 Comments: 1

1)∫(√(sin(x))) dx 2)∫cos(x^2 )dx

$$\left.\mathrm{1}\right)\int\sqrt{{sin}\left({x}\right)}\:{dx} \\ $$$$\left.\mathrm{2}\right)\int{cos}\left({x}^{\mathrm{2}} \right){dx} \\ $$$$ \\ $$

Question Number 84600 Answers: 0 Comments: 9

lim_(x→∞ ) x(√((x−1)/(9x+2))) − (x/3)

$$\underset{{x}\rightarrow\infty\:} {\mathrm{lim}}\:{x}\sqrt{\frac{{x}−\mathrm{1}}{\mathrm{9}{x}+\mathrm{2}}}\:−\:\frac{{x}}{\mathrm{3}} \\ $$

Question Number 84588 Answers: 2 Comments: 0

if x − (1/x) = 9 x^3 −(1/x^3 ) = ?

$$\mathrm{if}\:\mathrm{x}\:−\:\frac{\mathrm{1}}{\mathrm{x}}\:=\:\mathrm{9} \\ $$$$\mathrm{x}^{\mathrm{3}} −\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{3}} }\:=\:? \\ $$

Question Number 84582 Answers: 0 Comments: 10

Question Number 84581 Answers: 0 Comments: 2

let f(x) = e^(2x) ln(1−3x^2 ) 1) calculate f^((0)) (x) and f^((n)) (0) 2) drvelopp f at integr serie 3) find ∫ f(x)dx

$${let}\:{f}\left({x}\right)\:=\:{e}^{\mathrm{2}{x}} {ln}\left(\mathrm{1}−\mathrm{3}{x}^{\mathrm{2}} \right) \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left(\mathrm{0}\right)} \left({x}\right)\:{and}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:{drvelopp}\:{f}\:{at}\:{integr}\:{serie} \\ $$$$\left.\mathrm{3}\right)\:{find}\:\int\:{f}\left({x}\right){dx} \\ $$

Question Number 84580 Answers: 0 Comments: 0

find A_λ =∫ ((x+1)/(x+3))(√((λ+x)/(λ−x)))dx

$${find}\:{A}_{\lambda} =\int\:\frac{{x}+\mathrm{1}}{{x}+\mathrm{3}}\sqrt{\frac{\lambda+{x}}{\lambda−{x}}}{dx} \\ $$

Question Number 84579 Answers: 0 Comments: 0

find ∫ (x^2 −2)(√(x+(1/x)))dx

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

Question Number 84578 Answers: 0 Comments: 4

calculate ∫_0 ^(π/4) (dx/((cosx +3sinx)^2 ))

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{dx}}{\left({cosx}\:+\mathrm{3}{sinx}\right)^{\mathrm{2}} } \\ $$

Question Number 84577 Answers: 0 Comments: 2

calculate ∫ (dx/(cosx +cos(2x)+cos(3x)))

$${calculate}\:\int\:\:\:\:\frac{{dx}}{{cosx}\:+{cos}\left(\mathrm{2}{x}\right)+{cos}\left(\mathrm{3}{x}\right)} \\ $$

Question Number 84576 Answers: 0 Comments: 0

find locus of ∣z−(1/z)∣=2∣z^− ∣

$${find}\:{locus}\:{of}\:\:\:\mid{z}−\frac{\mathrm{1}}{{z}}\mid=\mathrm{2}\mid\overset{−} {{z}}\mid \\ $$

Question Number 84575 Answers: 0 Comments: 0

find nature of the serie Σ_(n=1) ^∞ Γ((1/n)) Γ(x)=∫_0 ^∞ t^(x−1) e^(−t) dt (x>0)

$${find}\:{nature}\:{of}\:{the}\:{serie}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\Gamma\left(\frac{\mathrm{1}}{{n}}\right) \\ $$$$\Gamma\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} \:{dt}\:\:\:\:\:\left({x}>\mathrm{0}\right) \\ $$

Question Number 84574 Answers: 0 Comments: 1

calculate I_n =∫_0 ^1 sin(narcsinx)dx

$${calculate}\:\:{I}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:{sin}\left({narcsinx}\right){dx} \\ $$

Question Number 84572 Answers: 0 Comments: 0

let F(z) =(z^2 /(1+z^7 )) 1) factorize inside C[x] and R[x] z^7 +1 2) decompose inside C(x)and R(x) the fraction F(x)

$${let}\:\:{F}\left({z}\right)\:=\frac{{z}^{\mathrm{2}} }{\mathrm{1}+{z}^{\mathrm{7}} } \\ $$$$\left.\mathrm{1}\right)\:{factorize}\:{inside}\:{C}\left[{x}\right]\:{and}\:{R}\left[{x}\right] \\ $$$${z}^{\mathrm{7}} \:+\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{decompose}\:{inside}\:{C}\left({x}\right){and}\:{R}\left({x}\right) \\ $$$${the}\:{fraction}\:{F}\left({x}\right) \\ $$

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