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Question Number 89946    Answers: 1   Comments: 0

∫ _(−(π/2)) ^(π/2) (dx/(1+e^(sin x) ))

$$\int\underset{−\frac{\pi}{\mathrm{2}}} {\overset{\frac{\pi}{\mathrm{2}}} {\:}}\:\frac{\mathrm{dx}}{\mathrm{1}+\mathrm{e}^{\mathrm{sin}\:\mathrm{x}} } \\ $$

Question Number 89958    Answers: 0   Comments: 1

prove that (1+sin x/1+cos 3(1sin x/1+cosec x)=tanx

$${prove}\:{that}\:\left(\mathrm{1}+\mathrm{sin}\:{x}/\mathrm{1}+\mathrm{cos}\:\mathrm{3}\left(\mathrm{1sin}\:{x}/\mathrm{1}+\mathrm{cosec}\:{x}\right)={tanx}\right. \\ $$

Question Number 89956    Answers: 0   Comments: 1

simplifyκgivingκyourκanswerκinκindexκform (√((ac^2 )/(9a^2 c^4 )))

$$\mathrm{simplify}\kappa\mathrm{giving}\kappa\mathrm{your}\kappa\mathrm{answer}\kappa\mathrm{in}\kappa\mathrm{index}\kappa\mathrm{form} \\ $$$$\sqrt{\frac{\mathrm{ac}^{\mathrm{2}} }{\mathrm{9a}^{\mathrm{2}} \mathrm{c}^{\mathrm{4}} }} \\ $$

Question Number 89938    Answers: 0   Comments: 1

If x(x+1) = 1 find (x+1)^3 −(1/((x+1)^3 ))

$$\mathrm{If}\:\mathrm{x}\left(\mathrm{x}+\mathrm{1}\right)\:=\:\mathrm{1}\: \\ $$$$\mathrm{find}\:\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{3}} −\frac{\mathrm{1}}{\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{3}} } \\ $$

Question Number 89937    Answers: 0   Comments: 1

Prove that for all complex such as ∣z∣<1= Σ_(n=1) ^∞ (z^n /((z^n −1)^2 )) +Σ_(n=1) ^∞ ((nz^n )/(z^n −1)) = 0

$${Prove}\:{that}\:{for}\:{all}\:{complex}\:{such}\:{as}\:\mid{z}\mid<\mathrm{1}= \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{z}^{{n}} }{\left({z}^{{n}} −\mathrm{1}\right)^{\mathrm{2}} }\:+\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{{nz}^{{n}} }{{z}^{{n}} −\mathrm{1}}\:=\:\mathrm{0}\: \\ $$

Question Number 89936    Answers: 1   Comments: 0

Prove that Σ_(p≥1,q≥1) (1/(pq(p+q−1))) =(π^2 /3)

$${Prove}\:{that}\:\underset{{p}\geqslant\mathrm{1},{q}\geqslant\mathrm{1}} {\sum}\:\:\frac{\mathrm{1}}{{pq}\left({p}+{q}−\mathrm{1}\right)}\:=\frac{\pi^{\mathrm{2}} }{\mathrm{3}}\: \\ $$

Question Number 89934    Answers: 0   Comments: 0

Let x∈]0;1[ Prove that Σ_(n=1) ^∞ (x^n /(1+x^n )) +Σ_(n=1) ^∞ (((−x)^n )/(1−x^n )) = 0

$$\left.{Let}\:{x}\in\right]\mathrm{0};\mathrm{1}\left[\:\:{Prove}\:{that}\right. \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{x}^{{n}} }{\mathrm{1}+{x}^{{n}} }\:+\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−{x}\right)^{{n}} }{\mathrm{1}−{x}^{{n}} }\:=\:\mathrm{0} \\ $$

Question Number 89928    Answers: 1   Comments: 0

Question Number 89925    Answers: 0   Comments: 0

x^2 (yy′′−y^2 )+xyy′ = y(√(x^2 (y′)^2 +y^2 ))

$${x}^{\mathrm{2}} \left({yy}''−{y}^{\mathrm{2}} \right)+{xyy}'\:=\:{y}\sqrt{{x}^{\mathrm{2}} \left({y}'\right)^{\mathrm{2}} +{y}^{\mathrm{2}} }\: \\ $$

Question Number 89922    Answers: 0   Comments: 2

Question Number 89918    Answers: 0   Comments: 1

∫ ((x tan^(−1) (x))/((1+x^2 )^(3/2) )) dx

$$\int\:\frac{\mathrm{x}\:\mathrm{tan}^{−\mathrm{1}} \left(\mathrm{x}\right)}{\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{3}/\mathrm{2}} }\:\mathrm{dx}\: \\ $$

Question Number 89913    Answers: 1   Comments: 1

Question Number 89908    Answers: 0   Comments: 6

Solve the differential equstion: (d^2 y/dx^2 ) = ((y_0 − 2y_(−1) + y_(−2) )/h^2 )

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{differential}\:\mathrm{equstion}: \\ $$$$\:\:\:\:\:\:\frac{\mathrm{d}^{\mathrm{2}} \mathrm{y}}{\mathrm{dx}^{\mathrm{2}} }\:\:\:=\:\:\:\frac{\mathrm{y}_{\mathrm{0}} \:\:−\:\:\mathrm{2y}_{−\mathrm{1}} \:\:+\:\:\mathrm{y}_{−\mathrm{2}} }{\mathrm{h}^{\mathrm{2}} } \\ $$

Question Number 89907    Answers: 1   Comments: 0

If the sum of 4 numbers is between 53 and 57 then the arithmetic mean of the numbers could be one of the following a)11.5 b)12 c)12.5 d)13 e)14

$${If}\:{the}\:{sum}\:{of}\:\mathrm{4}\:{numbers}\:{is}\:{between} \\ $$$$\mathrm{53}\:{and}\:\mathrm{57}\:{then}\:{the}\:{arithmetic}\:{mean}\:{of} \\ $$$${the}\:{numbers}\:{could}\:{be}\:{one}\:{of}\:{the} \\ $$$${following} \\ $$$$ \\ $$$$\left.{a}\left.\right)\left.\mathrm{1}\left.\mathrm{1}\left..\mathrm{5}\:{b}\right)\mathrm{12}\:{c}\right)\mathrm{12}.\mathrm{5}\:{d}\right)\mathrm{13}\:{e}\right)\mathrm{14} \\ $$

Question Number 89906    Answers: 1   Comments: 0

Show that 2x^7 −4x^4 +4x^2 =6x^6 +3 Has no solution in N

$${Show}\:{that}\: \\ $$$$\mathrm{2}{x}^{\mathrm{7}} −\mathrm{4}{x}^{\mathrm{4}} +\mathrm{4}{x}^{\mathrm{2}} =\mathrm{6}{x}^{\mathrm{6}} +\mathrm{3} \\ $$$${Has}\:{no}\:{solution}\:{in}\:\mathbb{N} \\ $$

Question Number 89898    Answers: 0   Comments: 3

In a classroom when the students sit 2 per bench, 11 students are left with no sits. And when they sit 3 per bench 7 benches are left empty. Determine the number of students in this classroom.

$${In}\:{a}\:{classroom}\:{when}\:{the}\:{students}\:{sit} \\ $$$$\mathrm{2}\:{per}\:{bench},\:\mathrm{11}\:{students}\:{are}\:{left}\:{with} \\ $$$${no}\:{sits}.\:{And}\:{when}\:{they}\:{sit}\:\mathrm{3}\:{per}\:{bench} \\ $$$$\mathrm{7}\:{benches}\:{are}\:{left}\:{empty}.\:{Determine}\:{the}\:{number} \\ $$$${of}\:{students}\:{in}\:{this}\:{classroom}. \\ $$

Question Number 89896    Answers: 0   Comments: 4

Question Number 89891    Answers: 0   Comments: 1

calculate ∫_(1/e) ^1 ln(x)ln(1+x)dx

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

Question Number 89882    Answers: 2   Comments: 0

Question Number 89878    Answers: 1   Comments: 1

A four_digit whole number is interesting if the number formed by the leftmost two digits is twice as large as the number formed by the rightmost two digits. (for example 2010 is interesting) 1 find the largest whole number B such that all interesting numbers are divisible by B 2 find the smallest whole number D such that D is divisible by all interesting numbers.

$$\boldsymbol{{A}}\:\boldsymbol{{four\_digit}}\:\boldsymbol{{whole}}\:\boldsymbol{{number}} \\ $$$$\boldsymbol{{is}}\:\boldsymbol{{interesting}}\:\boldsymbol{{if}}\:\boldsymbol{{the}}\:\boldsymbol{{number}} \\ $$$$\boldsymbol{{formed}}\:\boldsymbol{{by}}\:\boldsymbol{{the}}\:\boldsymbol{{leftmost}}\:\boldsymbol{{two}} \\ $$$$\boldsymbol{{digits}}\:\boldsymbol{{is}}\:\boldsymbol{{twice}}\:\boldsymbol{{as}}\:\boldsymbol{{large}}\:\boldsymbol{{as}}\:\boldsymbol{{the}} \\ $$$$\boldsymbol{{number}}\:\boldsymbol{{formed}}\:\boldsymbol{{by}}\:\boldsymbol{{the}} \\ $$$$\boldsymbol{{rightmost}}\:\boldsymbol{{two}}\:\boldsymbol{{digits}}. \\ $$$$\left(\boldsymbol{{for}}\:\boldsymbol{{example}}\:\mathrm{2010}\:\boldsymbol{{is}}\:\boldsymbol{{interesting}}\right) \\ $$$$\mathrm{1}\:\:\boldsymbol{{find}}\:\boldsymbol{{the}}\:\boldsymbol{{largest}}\:\boldsymbol{{whole}}\:\boldsymbol{{number}} \\ $$$$\mathbb{B}\:\boldsymbol{{such}}\:\boldsymbol{{that}}\:\boldsymbol{{all}}\:\boldsymbol{{interesting}} \\ $$$$\boldsymbol{{numbers}}\:\boldsymbol{{are}}\:\boldsymbol{{divisible}}\:\boldsymbol{{by}}\:\mathbb{B} \\ $$$$\mathrm{2}\:\:\boldsymbol{{find}}\:\boldsymbol{{the}}\:\boldsymbol{{smallest}}\:\boldsymbol{{whole}} \\ $$$$\boldsymbol{{number}}\:\mathbb{D}\:\boldsymbol{{such}}\:\boldsymbol{{that}}\:\mathbb{D}\:\boldsymbol{{is}} \\ $$$$\boldsymbol{{divisible}}\:\boldsymbol{{by}}\:\boldsymbol{{all}}\:\boldsymbol{{interesting}} \\ $$$$\boldsymbol{{numbers}}. \\ $$

Question Number 89874    Answers: 0   Comments: 0

∫_(1/3) ^(2/3) ((ln(1+(√(x^2 −(1/3)))))/(x(√(x^2 −(1/3)))))dx

$$\int_{\frac{\mathrm{1}}{\mathrm{3}}} ^{\frac{\mathrm{2}}{\mathrm{3}}} \:\frac{{ln}\left(\mathrm{1}+\sqrt{{x}^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{3}}}\right)}{{x}\sqrt{{x}^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{3}}}}{dx} \\ $$

Question Number 89852    Answers: 1   Comments: 0

Question Number 89849    Answers: 3   Comments: 1

Question Number 89848    Answers: 0   Comments: 1

Q1. If sinθ=(3/5) then find the value of tanθ+cotθ

$${Q}\mathrm{1}.\:{If}\:{sin}\theta=\frac{\mathrm{3}}{\mathrm{5}}\:{then}\:{find}\:{the}\:{value}\:{of}\:{tan}\theta+{cot}\theta \\ $$

Question Number 89846    Answers: 1   Comments: 3

∫_(−1) ^1 x cosh(x) ln(1+e^x ) dx

$$\int_{−\mathrm{1}} ^{\mathrm{1}} \:{x}\:{cosh}\left({x}\right)\:{ln}\left(\mathrm{1}+{e}^{{x}} \right)\:{dx} \\ $$

Question Number 89835    Answers: 2   Comments: 0

x^2 (d^2 y/dx^2 ) + 4x (dy/dx) + 2y = e^x

$${x}^{\mathrm{2}} \:\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\:+\:\mathrm{4}{x}\:\frac{{dy}}{{dx}}\:+\:\mathrm{2}{y}\:=\:{e}^{{x}} \\ $$

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