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Question Number 39458    Answers: 0   Comments: 4

find the greatest possible square insribed in a triangle with sides a b c

$$\mathrm{find}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{possible}\:\mathrm{square}\:\mathrm{insribed}\:\mathrm{in} \\ $$$$\mathrm{a}\:\mathrm{triangle}\:\mathrm{with}\:\mathrm{sides}\:{a}\:{b}\:{c} \\ $$

Question Number 39457    Answers: 1   Comments: 0

Question Number 39443    Answers: 1   Comments: 3

lim_(n→∞) [ (1/(n^2 +1))+ (2/(n^2 +2))+ (3/(n^2 +3))+ ....+(1/(n+1))] = ?

$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\left[\:\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{2}} +\mathrm{1}}+\:\frac{\mathrm{2}}{\mathrm{n}^{\mathrm{2}} +\mathrm{2}}+\:\frac{\mathrm{3}}{\mathrm{n}^{\mathrm{2}} +\mathrm{3}}+\:....+\frac{\mathrm{1}}{\mathrm{n}+\mathrm{1}}\right]\:=\:? \\ $$

Question Number 39441    Answers: 0   Comments: 2

∫_(1/4) ^( 4) (1/x) sin (x−(1/x))dx = ?

$$\int_{\frac{\mathrm{1}}{\mathrm{4}}} ^{\:\mathrm{4}} \:\frac{\mathrm{1}}{{x}}\:\mathrm{sin}\:\left({x}−\frac{\mathrm{1}}{{x}}\right){dx}\:=\:? \\ $$

Question Number 39440    Answers: 1   Comments: 0

f(x)= ∫_0 ^( x_ ) e^(t ) (((1+sin t)/(1+cos t))) dt. Then f((π/3))×f(((2π)/3)) = ?

$$\mathrm{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{\:{x}_{} } \:{e}^{{t}\:} \left(\frac{\mathrm{1}+\mathrm{sin}\:{t}}{\mathrm{1}+\mathrm{cos}\:{t}}\right)\:{dt}. \\ $$$${T}\mathrm{hen}\:\:\mathrm{f}\left(\frac{\pi}{\mathrm{3}}\right)×\mathrm{f}\left(\frac{\mathrm{2}\pi}{\mathrm{3}}\right)\:=\:? \\ $$

Question Number 39436    Answers: 1   Comments: 2

Question Number 39477    Answers: 1   Comments: 3

∫2^x 3^(2x) dx=?

$$\int\mathrm{2}^{\mathrm{x}} \mathrm{3}^{\mathrm{2x}} \mathrm{dx}=? \\ $$

Question Number 39431    Answers: 1   Comments: 1

∫_0 ^(2π) e^(x/2) sin ((x/2)+(π/4))dx = ?

$$\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\mathrm{e}^{\frac{{x}}{\mathrm{2}}} \mathrm{sin}\:\left(\frac{{x}}{\mathrm{2}}+\frac{\pi}{\mathrm{4}}\right)\mathrm{d}{x}\:=\:? \\ $$

Question Number 39402    Answers: 1   Comments: 1

Question Number 39483    Answers: 0   Comments: 3

find f(t)= ∫_0 ^1 ((ln(1+xt))/(1+x^2 )) dx .

$${find}\:{f}\left({t}\right)=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}\left(\mathrm{1}+{xt}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx}\:. \\ $$

Question Number 39395    Answers: 0   Comments: 2

Given that θ is an obtuse angle find tan θ if cos θ =(3/5)

$${Given}\:{that}\:\theta\:{is}\:{an}\:{obtuse}\: \\ $$$${angle}\:{find}\:{tan}\:\theta\:{if} \\ $$$${cos}\:\theta\:=\frac{\mathrm{3}}{\mathrm{5}} \\ $$$$ \\ $$

Question Number 39389    Answers: 0   Comments: 2

calculate F(x) = ∫_0 ^∞ (dt/(1+(1+x(1+t^2 ))^2 ))

$${calculate}\:{F}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\:\frac{{dt}}{\mathrm{1}+\left(\mathrm{1}+{x}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)\right)^{\mathrm{2}} } \\ $$

Question Number 39388    Answers: 1   Comments: 0

calculate A =tan((π/5)).tan(((2π)/5)).tan(((3π)/5)).tan(((4π)/5))

$${calculate}\:{A}\:={tan}\left(\frac{\pi}{\mathrm{5}}\right).{tan}\left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right).{tan}\left(\frac{\mathrm{3}\pi}{\mathrm{5}}\right).{tan}\left(\frac{\mathrm{4}\pi}{\mathrm{5}}\right) \\ $$

Question Number 39386    Answers: 1   Comments: 1

find the value of ∫_0 ^1 ((ln(1+x))/(1+x^2 ))dx

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

Question Number 39384    Answers: 2   Comments: 0

The values of a for which y= ax^2 +ax+(1/(24)) and x = ay^2 +ay+(1/(24)) touch each other are 1) (2/3) 2) (3/2) 3) ((13+(√(601)))/(12)) 4) ((13−(√(601)))/(12)).

$$\mathrm{The}\:\mathrm{values}\:\mathrm{of}\:\mathrm{a}\:\mathrm{for}\:\mathrm{which}\:\mathrm{y}=\:\mathrm{a}{x}^{\mathrm{2}} +{ax}+\frac{\mathrm{1}}{\mathrm{24}} \\ $$$${and}\:{x}\:=\:{ay}^{\mathrm{2}} +{ay}+\frac{\mathrm{1}}{\mathrm{24}}\:{touch}\:{each}\:{other} \\ $$$${are} \\ $$$$\left.\mathrm{1}\left.\right)\:\frac{\mathrm{2}}{\mathrm{3}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}\right)\:\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$\left.\mathrm{3}\left.\right)\:\frac{\mathrm{13}+\sqrt{\mathrm{601}}}{\mathrm{12}}\:\:\:\:\:\:\:\mathrm{4}\right)\:\frac{\mathrm{13}−\sqrt{\mathrm{601}}}{\mathrm{12}}. \\ $$

Question Number 39383    Answers: 1   Comments: 1

calculate ∫_0 ^(π/3) ((sinxdx)/(cosx(2+ln(cosx))) .

$${calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{3}}} \:\:\:\:\:\:\frac{{sinxdx}}{{cosx}\left(\mathrm{2}+{ln}\left({cosx}\right)\right.}\:. \\ $$

Question Number 39382    Answers: 1   Comments: 0

Question Number 39381    Answers: 1   Comments: 0

Question Number 39380    Answers: 0   Comments: 0

find the value of Σ_(n=0) ^∞ (((−1)^n )/(4n+1))

$${find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\mathrm{4}{n}+\mathrm{1}} \\ $$

Question Number 39379    Answers: 1   Comments: 6

Question Number 39378    Answers: 0   Comments: 1

study the derivability of f(x)=Σ_(n=0) ^∞ (((−1)^n )/(nx +1))

$${study}\:{the}\:{derivability}\:{of} \\ $$$${f}\left({x}\right)=\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{nx}\:+\mathrm{1}} \\ $$

Question Number 39376    Answers: 0   Comments: 0

how to calculate the product (Σ_(n=0) ^∞ a_n x^n ).(Σ_(n=0) ^∞ b_n x^(2n) )?

$${how}\:{to}\:{calculate}\:{the}\:{product}\:\left(\sum_{{n}=\mathrm{0}} ^{\infty} {a}_{{n}} {x}^{{n}} \right).\left(\sum_{{n}=\mathrm{0}} ^{\infty} \:{b}_{{n}} \:{x}^{\mathrm{2}{n}} \right)? \\ $$

Question Number 39375    Answers: 0   Comments: 1

Question Number 39374    Answers: 0   Comments: 2

calculate I = ∫_0 ^∞ ((arctan(x^2 ))/(1+x^2 ))dx

$${calculate}\:{I}\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{arctan}\left({x}^{\mathrm{2}} \right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 39373    Answers: 0   Comments: 1

find the values of integrals A = ∫_(−∞) ^(+∞) cos(x^2 +x+1)dx and B = ∫_(−∞) ^(+∞) sin(x^2 +x+1)dx

$${find}\:{the}\:{values}\:{of}\:{integrals} \\ $$$${A}\:=\:\int_{−\infty} ^{+\infty} \:{cos}\left({x}^{\mathrm{2}} \:+{x}+\mathrm{1}\right){dx}\:\:\:{and}\:{B}\:=\:\int_{−\infty} ^{+\infty} \:{sin}\left({x}^{\mathrm{2}} \:+{x}+\mathrm{1}\right){dx} \\ $$

Question Number 39371    Answers: 2   Comments: 1

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