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Question Number 137873    Answers: 0   Comments: 3

.......nice ............calculus....... 𝛗=Ξ£_(n=1) ^∞ ((sin(nx))/n) =(Ο€/2)βˆ’(x/2) 𝛗=βˆ’Im(1βˆ’e^(ix) )=βˆ’Imln{(1βˆ’cos(x)βˆ’isin(x))} =βˆ’Im{ln((√((1βˆ’cos(x))^2 +sin^2 (x))) +itan^(βˆ’1) (((βˆ’sin(x))/(1βˆ’cos(x))))} =βˆ’Im{(√(2βˆ’2cos(x)_ )) βˆ’itan^(βˆ’1) (tan((Ο€/2)βˆ’(x/2)))} =tan^(βˆ’1) (tan((Ο€/2)βˆ’(x/2))) ........ 𝛗=(Ο€/2)βˆ’(x/2) ...βœ“.....βŸ¨βˆ—βŸ© ^(β€²β€²) ∫ ^(β€²β€²) both sides of βŸ¨βˆ—βŸ© βˆ’Ξ£_(n=1) ^∞ ((cos(nx))/n^2 )dx+C=((Ο€x)/2)βˆ’(x^2 /4) x=0 β‡’ C=(Ο€^2 /6) ...βœ“ Ξ£_(n=1) ^∞ ((cos(nx))/n^2 )=βˆ’((Ο€x)/2)+(x^2 /4)+(Ο€^2 /6) ....βœ“ x=Ο€(1+(√(1/3)) ) .... Ξ£((cos(nΟ€(1+(√(1/3)) )))/n^2 )=(1/2)(βˆ’Ο€^2 (1+(√(1/3)) )+((Ο€^2 (1+(√(1/3)) )^2 )/2)+(Ο€^2 /3)) =(Ο€^2 /2)(βˆ’1βˆ’((√3)/3) +((1+2.((√3)/3) +(1/3))/2)+(1/3)) =(Ο€^2 /2)(βˆ’1βˆ’(((√3) )/3)+(4/6)+(((√3) )/3)+(1/3))=0..βœ“βœ“ ......prepared by mr (Dwaipayan)..... m.n#

$$\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:.......{nice}\:\:............{calculus}....... \\ $$$$\:\:\:\:\boldsymbol{\phi}=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{sin}\left({nx}\right)}{{n}}\:=\frac{\pi}{\mathrm{2}}βˆ’\frac{{x}}{\mathrm{2}} \\ $$$$\:\:\:\boldsymbol{\phi}=βˆ’{Im}\left(\mathrm{1}βˆ’{e}^{{ix}} \right)=βˆ’{Imln}\left\{\left(\mathrm{1}βˆ’{cos}\left({x}\right)βˆ’{isin}\left({x}\right)\right)\right\} \\ $$$$\:\:\:\:=βˆ’{Im}\left\{{ln}\left(\sqrt{\left(\mathrm{1}βˆ’{cos}\left({x}\right)\right)^{\mathrm{2}} +{sin}^{\mathrm{2}} \left({x}\right)}\:+{itan}^{βˆ’\mathrm{1}} \left(\frac{βˆ’{sin}\left({x}\right)}{\mathrm{1}βˆ’{cos}\left({x}\right)}\right)\right\}\right. \\ $$$$=βˆ’{Im}\left\{\sqrt{\mathrm{2}βˆ’\mathrm{2}{cos}\left({x}\right)_{\:} }\:βˆ’{itan}^{βˆ’\mathrm{1}} \left({tan}\left(\frac{\pi}{\mathrm{2}}βˆ’\frac{{x}}{\mathrm{2}}\right)\right)\right\} \\ $$$$\:\:={tan}^{βˆ’\mathrm{1}} \left({tan}\left(\frac{\pi}{\mathrm{2}}βˆ’\frac{{x}}{\mathrm{2}}\right)\right)\:\: \\ $$$$\:\:\:\:\:\:\:........\:\:\:\boldsymbol{\phi}=\frac{\pi}{\mathrm{2}}βˆ’\frac{{x}}{\mathrm{2}}\:\:...\checkmark.....\langle\ast\rangle \\ $$$$\:\:\:\:\:^{''} \:\:\int\:\:^{''} \:\:{both}\:{sides}\:{of}\:\langle\ast\rangle \\ $$$$\:\:\:\:\:\:βˆ’\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{{cos}\left({nx}\right)}{{n}^{\mathrm{2}} }{dx}+{C}=\frac{\pi{x}}{\mathrm{2}}βˆ’\frac{{x}^{\mathrm{2}} }{\mathrm{4}} \\ $$$$\:\:\:\:\:{x}=\mathrm{0}\:\Rightarrow\:{C}=\frac{\pi^{\mathrm{2}} }{\mathrm{6}}\:\:\:\:...\checkmark \\ $$$$\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{cos}\left({nx}\right)}{{n}^{\mathrm{2}} }=βˆ’\frac{\pi{x}}{\mathrm{2}}+\frac{{x}^{\mathrm{2}} }{\mathrm{4}}+\frac{\pi^{\mathrm{2}} }{\mathrm{6}}\:....\checkmark \\ $$$$\:\:\:\:\:\:\:\:{x}=\pi\left(\mathrm{1}+\sqrt{\frac{\mathrm{1}}{\mathrm{3}}}\:\right)\:.... \\ $$$$\:\:\:\:\:\:\:\:\Sigma\frac{{cos}\left({n}\pi\left(\mathrm{1}+\sqrt{\frac{\mathrm{1}}{\mathrm{3}}}\:\right)\right)}{{n}^{\mathrm{2}} }=\frac{\mathrm{1}}{\mathrm{2}}\left(βˆ’\pi^{\mathrm{2}} \left(\mathrm{1}+\sqrt{\frac{\mathrm{1}}{\mathrm{3}}}\:\right)+\frac{\pi^{\mathrm{2}} \left(\mathrm{1}+\sqrt{\frac{\mathrm{1}}{\mathrm{3}}}\:\right)^{\mathrm{2}} }{\mathrm{2}}+\frac{\pi^{\mathrm{2}} }{\mathrm{3}}\right) \\ $$$$=\frac{\pi^{\mathrm{2}} }{\mathrm{2}}\left(βˆ’\mathrm{1}βˆ’\frac{\sqrt{\mathrm{3}}}{\mathrm{3}}\:+\frac{\mathrm{1}+\mathrm{2}.\frac{\sqrt{\mathrm{3}}}{\mathrm{3}}\:+\frac{\mathrm{1}}{\mathrm{3}}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}\right) \\ $$$$=\frac{\pi^{\mathrm{2}} }{\mathrm{2}}\left(βˆ’\mathrm{1}βˆ’\frac{\sqrt{\mathrm{3}}\:}{\mathrm{3}}+\frac{\mathrm{4}}{\mathrm{6}}+\frac{\sqrt{\mathrm{3}}\:}{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{3}}\right)=\mathrm{0}..\checkmark\checkmark \\ $$$$\:\:\:\:\:\:\:\:\:\:......{prepared}\:{by}\:{mr}\:\left({Dwaipayan}\right)..... \\ $$$$\:\:{m}.{n}# \\ $$$$\:\:\:\: \\ $$

Question Number 137841    Answers: 0   Comments: 1

Question Number 137838    Answers: 1   Comments: 1

∫log_x edx=??

$$\int{log}_{{x}} {edx}=?? \\ $$

Question Number 137837    Answers: 4   Comments: 0

∫_0 ^∞ ((sin xβˆ’sin x^2 )/x)=(Ο€/4) ∫_0 ^∞ ((cos xβˆ’cos x^2 )/x)dx=βˆ’(Ξ³/2)

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{sin}\:{x}βˆ’\mathrm{sin}\:{x}^{\mathrm{2}} }{{x}}=\frac{\pi}{\mathrm{4}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{cos}\:{x}βˆ’\mathrm{cos}\:{x}^{\mathrm{2}} }{{x}}{dx}=βˆ’\frac{\gamma}{\mathrm{2}} \\ $$

Question Number 137836    Answers: 0   Comments: 0

Question Number 137829    Answers: 1   Comments: 0

.......nice ... ... .... calculus..... prove that :::: 𝛗=∫_0 ^( 1) (((log(1βˆ’x))/x))^2 dx=2ΞΆ(2)....

$$\:\:\:\:\:.......{nice}\:\:...\:...\:....\:{calculus}..... \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:{prove}\:{that}\::::: \\ $$$$\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\mathrm{1}} \left(\frac{{log}\left(\mathrm{1}βˆ’{x}\right)}{{x}}\right)^{\mathrm{2}} {dx}=\mathrm{2}\zeta\left(\mathrm{2}\right).... \\ $$$$ \\ $$

Question Number 137820    Answers: 2   Comments: 0

lim_(xβ†’Ο€/2) (((1βˆ’tan (x/2))(1βˆ’sin x))/((1+tan (x/2))(Ο€βˆ’2x)^3 )) ?

$$\underset{{x}\rightarrow\pi/\mathrm{2}} {\mathrm{lim}}\:\frac{\left(\mathrm{1}βˆ’\mathrm{tan}\:\frac{{x}}{\mathrm{2}}\right)\left(\mathrm{1}βˆ’\mathrm{sin}\:{x}\right)}{\left(\mathrm{1}+\mathrm{tan}\:\frac{{x}}{\mathrm{2}}\right)\left(\piβˆ’\mathrm{2}{x}\right)^{\mathrm{3}} }\:? \\ $$

Question Number 137866    Answers: 0   Comments: 0

Question Number 137817    Answers: 1   Comments: 0

Let f(x)=x^2 βˆ’2xβˆ’3; xβ‰₯1 & g(x)=1+(√(x+4)) ; xβ‰₯βˆ’4 then the number of real solutions of equation f(x)=g(x) is ...

$${Let}\:{f}\left({x}\right)={x}^{\mathrm{2}} βˆ’\mathrm{2}{x}βˆ’\mathrm{3};\:{x}\geqslant\mathrm{1}\:\& \\ $$$${g}\left({x}\right)=\mathrm{1}+\sqrt{{x}+\mathrm{4}}\:;\:{x}\geqslantβˆ’\mathrm{4}\:{then} \\ $$$${the}\:{number}\:{of}\:{real}\:{solutions} \\ $$$${of}\:{equation}\:{f}\left({x}\right)={g}\left({x}\right)\:{is}\:... \\ $$

Question Number 137813    Answers: 1   Comments: 0

cos (arctan (((21)/(60))))=?

$$\mathrm{cos}\:\left(\mathrm{arctan}\:\left(\frac{\mathrm{21}}{\mathrm{60}}\right)\right)=? \\ $$

Question Number 137811    Answers: 2   Comments: 0

Given { ((cos (xβˆ’y)=βˆ’1+(1/2)cos x)),((cos (x+y)= 1+(1/3)cos x)) :} where 270Β° < y< 360Β° . find sin 2y .

$${Given}\:\begin{cases}{\mathrm{cos}\:\left({x}βˆ’{y}\right)=βˆ’\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}\:{x}}\\{\mathrm{cos}\:\left({x}+{y}\right)=\:\mathrm{1}+\frac{\mathrm{1}}{\mathrm{3}}\mathrm{cos}\:{x}}\end{cases} \\ $$$${where}\:\mathrm{270}Β°\:<\:{y}<\:\mathrm{360}Β°\:.\:{find} \\ $$$$\mathrm{sin}\:\mathrm{2}{y}\:. \\ $$

Question Number 137799    Answers: 2   Comments: 0

..... mathematical .. ... ... analysis.... evaluate :: 𝛗=∫_0 ^( 1) (((ln^2 (1βˆ’x^2 ))/x))=?

$$\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:.....\:{mathematical}\:..\:...\:...\:{analysis}.... \\ $$$$\:\:\:\:\:\:\:{evaluate}\:::\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\mathrm{1}} \left(\frac{{ln}^{\mathrm{2}} \left(\mathrm{1}βˆ’{x}^{\mathrm{2}} \right)}{{x}}\right)=? \\ $$$$ \\ $$

Question Number 137787    Answers: 1   Comments: 0

A string AB of lenght 2l has a particle attached to its midpoint C. The ends A and B of the string are fastened to two fixed points with A distance l vertically above B.With both parts of the string taunt,the particle describes a horizontal circle about the line AB with constant angular speed Ο‰. If the tension in CA is three times that in CB, prove that the angular velocity is 2(√(g/l)) .

$$\mathrm{A}\:\mathrm{string}\:{AB}\:\mathrm{of}\:\mathrm{lenght}\:\mathrm{2}{l}\:\mathrm{has}\:\mathrm{a}\:\mathrm{particle}\:\mathrm{attached}\:\mathrm{to}\:\mathrm{its}\:\mathrm{midpoint}\:\mathrm{C}.\: \\ $$$$\mathrm{The}\:\mathrm{ends}\:{A}\:\mathrm{and}\:{B}\:\mathrm{of}\:\mathrm{the}\:\mathrm{string}\:\mathrm{are}\:\mathrm{fastened}\:\mathrm{to}\:\mathrm{two}\:\mathrm{fixed}\:\mathrm{points} \\ $$$$\mathrm{with}\:{A}\:\mathrm{distance}\:{l}\:\mathrm{vertically}\:\mathrm{above}\:\mathrm{B}.\mathrm{With}\:\mathrm{both}\:\mathrm{parts}\:\mathrm{of}\:\mathrm{the}\:\mathrm{string} \\ $$$$\mathrm{taunt},\mathrm{the}\:\mathrm{particle}\:\mathrm{describes}\:\mathrm{a}\:\mathrm{horizontal}\:\mathrm{circle}\:\mathrm{about}\:\mathrm{the}\:\mathrm{line}\:{AB}\:\mathrm{with} \\ $$$$\mathrm{constant}\:\mathrm{angular}\:\mathrm{speed}\:\omega.\:\mathrm{If}\:\mathrm{the}\:\mathrm{tension}\:\mathrm{in}\:{CA}\:\mathrm{is}\:\mathrm{three}\:\mathrm{times} \\ $$$$\mathrm{that}\:\mathrm{in}\:{CB},\:\mathrm{prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{angular}\:\mathrm{velocity}\:\mathrm{is}\:\mathrm{2}\sqrt{\frac{\mathrm{g}}{{l}}}\:. \\ $$

Question Number 137784    Answers: 2   Comments: 0

∫_0 ^( (Ο€/6)) (dx/(sinx))=?

$$\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{6}}} \frac{{dx}}{{sinx}}=? \\ $$$$ \\ $$

Question Number 137783    Answers: 1   Comments: 0

Some birds were flying and met a bird, the bird greeted them and said how are you hundred. The birds said we are not hundred, we need half of us plus you to make hundred. How many birds were flying

$$\mathrm{Some}\:\mathrm{birds}\:\mathrm{were}\:\mathrm{flying}\:\mathrm{and}\:\mathrm{met}\:\mathrm{a}\: \\ $$$$\mathrm{bird},\:\mathrm{the}\:\mathrm{bird}\:\mathrm{greeted}\:\mathrm{them}\:\mathrm{and}\:\mathrm{said} \\ $$$$\mathrm{how}\:\mathrm{are}\:\mathrm{you}\:\mathrm{hundred}.\:\mathrm{The}\:\mathrm{birds}\:\mathrm{said} \\ $$$$\mathrm{we}\:\mathrm{are}\:\mathrm{not}\:\mathrm{hundred},\:\mathrm{we}\:\mathrm{need}\:\mathrm{half}\:\mathrm{of}\:\mathrm{us} \\ $$$$\mathrm{plus}\:\mathrm{you}\:\mathrm{to}\:\mathrm{make}\:\mathrm{hundred}.\:\mathrm{How}\:\mathrm{many} \\ $$$$\mathrm{birds}\:\mathrm{were}\:\mathrm{flying} \\ $$

Question Number 137902    Answers: 2   Comments: 0

lim_(xβ†’0) ((xβˆ’sin x)/(x^2 (√x))) =?

$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}βˆ’\mathrm{sin}\:{x}}{{x}^{\mathrm{2}} \:\sqrt{{x}}}\:=? \\ $$

Question Number 137773    Answers: 2   Comments: 0

lim_(xβ†’0) ((cos pxβˆ’cos qx)/x^2 )

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{cos}\:{px}βˆ’\mathrm{cos}\:{qx}}{{x}^{\mathrm{2}} } \\ $$

Question Number 137772    Answers: 2   Comments: 0

Question Number 137771    Answers: 0   Comments: 2

Question Number 137770    Answers: 1   Comments: 0

Question Number 137768    Answers: 1   Comments: 1

Question Number 137764    Answers: 1   Comments: 0

If log_2 3=a and log_3 7=b, express log_(42) 56 in terms of a and b

$$\mathrm{If}\:\mathrm{log}_{\mathrm{2}} \mathrm{3}={a}\:\mathrm{and}\:\mathrm{log}_{\mathrm{3}} \mathrm{7}={b},\:\mathrm{express} \\ $$$$\mathrm{log}_{\mathrm{42}} \mathrm{56}\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:{a}\:\mathrm{and}\:{b} \\ $$

Question Number 137751    Answers: 0   Comments: 0

Question Number 137750    Answers: 0   Comments: 2

Question Number 137746    Answers: 1   Comments: 0

Question Number 137745    Answers: 1   Comments: 0

Each of the digits 2, 4, 6, and 8 can be used once and once only in writing a four-digit number. What is the sum of all such numbers that are divisible by 11?

$$ \\ $$Each of the digits 2, 4, 6, and 8 can be used once and once only in writing a four-digit number. What is the sum of all such numbers that are divisible by 11?

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