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Question Number 217359 Answers: 2 Comments: 0

Find all three-digit numbers such that when the number is divided by the sum of its digits the quotient is 7 and the remainder is 5.

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{three}-\mathrm{digit}\:\mathrm{numbers}\:\mathrm{such}\:\mathrm{that}\:\mathrm{when}\:\mathrm{the}\:\mathrm{number}\:\mathrm{is} \\ $$$$\mathrm{divided}\:\mathrm{by}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{its}\:\mathrm{digits}\:\mathrm{the}\:\mathrm{quotient}\:\mathrm{is}\:\mathrm{7}\:\mathrm{and}\:\mathrm{the}\: \\ $$$$\mathrm{remainder}\:\mathrm{is}\:\mathrm{5}. \\ $$

Question Number 217358 Answers: 1 Comments: 0

Question Number 217356 Answers: 1 Comments: 0

Question Number 217326 Answers: 3 Comments: 0

Find three consecutive integers such that the sum of their squares is 50.

$${Find}\:{three}\:{consecutive}\:{integers}\: \\ $$$${such}\:{that}\:{the}\:{sum}\:{of}\:{their}\:{squares} \\ $$$$\:{is}\:\mathrm{50}. \\ $$

Question Number 217321 Answers: 2 Comments: 0

Find three consecutive integers such that the product of the first two is 16 more than twice the third integer. Provide all possible solutions.

$$\mathrm{Find}\:\mathrm{three}\:\mathrm{consecutive}\:\mathrm{integers}\:\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:\mathrm{product}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{first}\:\mathrm{two}\:\mathrm{is}\:\mathrm{16}\:\mathrm{more}\:\mathrm{than}\:\mathrm{twice}\:\mathrm{the}\:\mathrm{third}\:\mathrm{integer}.\:\mathrm{Provide}\:\mathrm{all}\: \\ $$$$\mathrm{possible}\:\mathrm{solutions}. \\ $$

Question Number 217314 Answers: 1 Comments: 9

How is ψ^2 (1) = − 2ζ(3) and ψ^2 ((1/2)) = − 14ζ(3) ???

$$\mathrm{How}\:\mathrm{is}\:\:\:\psi^{\mathrm{2}} \left(\mathrm{1}\right)\:\:=\:\:−\:\:\mathrm{2}\zeta\left(\mathrm{3}\right) \\ $$$$\:\:\:\:\:\:\mathrm{and}\:\:\:\:\psi^{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{2}}\right)\:\:\:=\:\:\:−\:\:\mathrm{14}\zeta\left(\mathrm{3}\right)\:\:\:\:\:\:\:??? \\ $$

Question Number 217300 Answers: 2 Comments: 0

Question Number 217299 Answers: 1 Comments: 0

Question Number 217291 Answers: 5 Comments: 0

Solve: ((x+2)/(x−3))−((x−1)/(x+4))=((10)/(x^2 +x−12))

$$\mathrm{Solve}: \\ $$$$\frac{{x}+\mathrm{2}}{{x}−\mathrm{3}}−\frac{{x}−\mathrm{1}}{{x}+\mathrm{4}}=\frac{\mathrm{10}}{{x}^{\mathrm{2}} +{x}−\mathrm{12}} \\ $$

Question Number 217290 Answers: 4 Comments: 0

∫_0 ^( 1) ((ln^2 (1−x))/x^2 ) dx= ?

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{ln}^{\mathrm{2}} \left(\mathrm{1}−{x}\right)}{{x}^{\mathrm{2}} }\:{dx}=\:? \\ $$$$ \\ $$$$ \\ $$

Question Number 217289 Answers: 1 Comments: 1

∫_0 ^1 sin^(−1) (1/( (√(1+x−x^2 )))) dx=?

$$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\mathrm{sin}^{−\mathrm{1}} \:\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+{x}−{x}^{\mathrm{2}} }}\:{dx}=? \\ $$

Question Number 217278 Answers: 0 Comments: 0

Question Number 217256 Answers: 1 Comments: 0

Question Number 217255 Answers: 1 Comments: 0

∫_(−∞) ^(+∞) e^(−(x^2 /2)) dx=(√(2π)),∫_(−∞) ^(+∞) e^(−(x^2 /2)+x) dx.

$$\int_{−\infty} ^{+\infty} {e}^{−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}} {dx}=\sqrt{\mathrm{2}\pi},\int_{−\infty} ^{+\infty} {e}^{−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}+{x}} {dx}. \\ $$

Question Number 217252 Answers: 3 Comments: 5

Question Number 217245 Answers: 1 Comments: 0

find the following differential equation by eliminating the arbritrary constant (1)y=Ae^x +Bcosx (2) xy=Ae^x +Be^(−x) +x^2

$${find}\:{the}\:{following}\:{differential}\:{equation}\: \\ $$$${by}\:{eliminating}\:{the}\:{arbritrary}\:{constant} \\ $$$$\left(\mathrm{1}\right){y}={Ae}^{{x}} +{Bcosx} \\ $$$$\left(\mathrm{2}\right)\:{xy}={Ae}^{{x}} +{Be}^{−{x}} +{x}^{\mathrm{2}} \\ $$$$ \\ $$

Question Number 217238 Answers: 2 Comments: 0

Question Number 217235 Answers: 3 Comments: 0

∫_( 0) ^( 1) ((x ln^2 (x))/(1 + x^2 )) dx

$$\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{x}\:\mathrm{ln}^{\mathrm{2}} \left(\mathrm{x}\right)}{\mathrm{1}\:\:+\:\:\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx} \\ $$

Question Number 217228 Answers: 3 Comments: 0

Question Number 217225 Answers: 1 Comments: 0

Find: ∫ − (dx/( (√(24x − 16x^2 − 8)))) = ?

$$\mathrm{Find}:\:\:\:\:\:\int\:−\:\frac{{d}\mathrm{x}}{\:\sqrt{\mathrm{24x}\:−\:\mathrm{16x}^{\mathrm{2}} \:−\:\mathrm{8}}}\:=\:? \\ $$

Question Number 217219 Answers: 1 Comments: 0

calculate determinant ((( L ( ∫_1 ^( ∞) (( e^( −tx) )/x)dx ) =_(transfom) ^(laplace) ? ; t>0 )))

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{calculate} \\ $$$$\begin{array}{|c|}{\:\:\mathscr{L}\:\:\left(\:\int_{\mathrm{1}} ^{\:\infty} \frac{\:\mathrm{e}^{\:−{tx}} }{{x}}{dx}\:\right)\:\underset{\mathrm{transfom}} {\overset{\mathrm{laplace}} {=}}?\:\:;\:\:{t}>\mathrm{0}\:}\\\hline\end{array} \\ $$$$ \\ $$$$\:\:\: \\ $$

Question Number 217211 Answers: 1 Comments: 0

a nice one: prove ∫_0 ^1 (√(−((ln t)/t))) dt=(√(2π))

$$\mathrm{a}\:\mathrm{nice}\:\mathrm{one}: \\ $$$$\mathrm{prove}\:\:\:\:\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\sqrt{−\frac{\mathrm{ln}\:{t}}{{t}}}\:{dt}=\sqrt{\mathrm{2}\pi} \\ $$

Question Number 217205 Answers: 1 Comments: 0

Question Number 217203 Answers: 2 Comments: 0

A farmer has 100 meters of fencing and wants to enclose an rectagular field along a river. Thei rver forms one side of the rectangle so fencing is needed onlyo for the other three sides. What dimesions should the farmer chooseto maximize the enclosed area?

$$\mathrm{A}\:\mathrm{farmer}\:\mathrm{has}\:\mathrm{100}\:\mathrm{meters}\:\mathrm{of}\: \\ $$$$\mathrm{fencing}\:\mathrm{and}\:\mathrm{wants}\:\mathrm{to}\:\mathrm{enclose}\:\mathrm{an} \\ $$$$\mathrm{rectagular}\:\mathrm{field}\:\mathrm{along}\:\mathrm{a}\:\mathrm{river}.\:\mathrm{Thei} \\ $$$$\mathrm{rver}\:\mathrm{forms}\:\mathrm{one}\:\mathrm{side}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{rectangle}\:\mathrm{so}\:\mathrm{fencing}\:\mathrm{is}\:\mathrm{needed}\:\mathrm{onlyo} \\ $$$$\mathrm{for}\:\mathrm{the}\:\mathrm{other}\:\mathrm{three}\:\mathrm{sides}.\:\mathrm{What}\: \\ $$$$\mathrm{dimesions}\:\mathrm{should}\:\mathrm{the}\:\mathrm{farmer}\: \\ $$$$\mathrm{chooseto}\:\mathrm{maximize}\:\mathrm{the}\:\mathrm{enclosed} \\ $$$$\mathrm{area}? \\ $$

Question Number 217199 Answers: 2 Comments: 1

Question Number 217198 Answers: 1 Comments: 0

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