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Question Number 84733 Answers: 1 Comments: 0

∫((x^4 +1)/(x^6 +1))dx

$$\int\frac{{x}^{\mathrm{4}} +\mathrm{1}}{{x}^{\mathrm{6}} +\mathrm{1}}{dx} \\ $$

Question Number 84728 Answers: 0 Comments: 8

y=x[x[x]] with x∈R^+ find the range of function and solve x[x[x]]=150.

$${y}={x}\left[{x}\left[{x}\right]\right]\:{with}\:{x}\in{R}^{+} \\ $$$${find}\:{the}\:{range}\:{of}\:{function} \\ $$$${and}\:{solve}\:{x}\left[{x}\left[{x}\right]\right]=\mathrm{150}. \\ $$

Question Number 84726 Answers: 0 Comments: 0

prove that ∫_0 ^1 (y^y )^((y^y )^((y^y )^.^.^. ) ) dy=(π^2 /(12))

$${prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \left({y}^{{y}} \right)^{\left({y}^{{y}} \right)^{\left({y}^{{y}} \right)^{.^{.^{.} } } } } {dy}=\frac{\pi^{\mathrm{2}} }{\mathrm{12}} \\ $$

Question Number 84718 Answers: 2 Comments: 0

Question Number 84785 Answers: 1 Comments: 2

at what time is the short clock and long hour hand form an angle of 180 degrees?

$$\mathrm{at}\:\mathrm{what}\:\mathrm{time}\:\mathrm{is}\:\mathrm{the}\:\mathrm{short}\:\mathrm{clock}\: \\ $$$$\mathrm{and}\:\mathrm{long}\:\mathrm{hour}\:\mathrm{hand}\:\mathrm{form}\:\mathrm{an}\: \\ $$$$\mathrm{angle}\:\mathrm{of}\:\mathrm{180}\:\mathrm{degrees}? \\ $$

Question Number 84713 Answers: 2 Comments: 2

∫ _0 ^( 1) cos^(−1) (x^2 −x+1) dx ?

$$\int\overset{\:\:\mathrm{1}} {\:}_{\mathrm{0}} \mathrm{cos}^{−\mathrm{1}} \left(\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{1}\right)\:\mathrm{dx}\:? \\ $$

Question Number 84711 Answers: 0 Comments: 2

Question Number 84702 Answers: 0 Comments: 3

Question Number 84709 Answers: 1 Comments: 0

∫((tan(x)))^(1/3) dx

$$\int\sqrt[{\mathrm{3}}]{{tan}\left({x}\right)}\:{dx} \\ $$

Question Number 84708 Answers: 1 Comments: 0

Question Number 84693 Answers: 1 Comments: 1

Question Number 84689 Answers: 1 Comments: 2

Question Number 84685 Answers: 0 Comments: 0

∫(dx/(x(2−e^x )))

$$\int\frac{{dx}}{{x}\left(\mathrm{2}−{e}^{{x}} \right)} \\ $$

Question Number 84683 Answers: 1 Comments: 0

x∈N and y∈N:(x<y) slove (E): xy−4y+3x−2027=0.....(E)

$${x}\in{N}\:{and}\:{y}\in{N}:\left({x}<{y}\right) \\ $$$${slove}\:\left({E}\right): \\ $$$${xy}−\mathrm{4}{y}+\mathrm{3}{x}−\mathrm{2027}=\mathrm{0}.....\left({E}\right) \\ $$

Question Number 84682 Answers: 0 Comments: 5

Question Number 84681 Answers: 1 Comments: 1

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

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