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Question Number 143477 Answers: 0 Comments: 0

∫_0 ^1 e^(2arctg(t^2 )) dt

$$\int_{\mathrm{0}} ^{\mathrm{1}} {e}^{\mathrm{2}{arctg}\left({t}^{\mathrm{2}} \right)} {dt} \\ $$

Question Number 143475 Answers: 1 Comments: 1

evaluate; ((((√7))^(log64) −(3)^(log_(24) 8) )/((log _2 8−log _(1/4) 64)((1/(log _4 ((1/(64))))))))

$$\:{evaluate}; \\ $$$$\frac{\left(\sqrt{\mathrm{7}}\right)^{\mathrm{log64}} −\left(\mathrm{3}\right)^{\mathrm{log}_{\mathrm{24}} \mathrm{8}} }{\left(\mathrm{log}\:_{\mathrm{2}} \mathrm{8}−\mathrm{log}\:_{\frac{\mathrm{1}}{\mathrm{4}}} \mathrm{64}\right)\left(\frac{\mathrm{1}}{\mathrm{log}\:_{\mathrm{4}} \left(\frac{\mathrm{1}}{\mathrm{64}}\right)}\right)} \\ $$

Question Number 143463 Answers: 0 Comments: 0

Π_(k=1) ^n tan(((kπ)/(2n+1)))=(√(2n+1))

$$\underset{{k}=\mathrm{1}} {\overset{{n}} {\prod}}{tan}\left(\frac{{k}\pi}{\mathrm{2}{n}+\mathrm{1}}\right)=\sqrt{\mathrm{2}{n}+\mathrm{1}} \\ $$

Question Number 143462 Answers: 3 Comments: 0

lim_(x→1) ((x−1)/(ln((x/(2−x))))) = ???

$$\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\:\:\frac{{x}−\mathrm{1}}{{ln}\left(\frac{{x}}{\mathrm{2}−{x}}\right)}\:=\:??? \\ $$

Question Number 143461 Answers: 3 Comments: 0

Prove that : ∀n∈N^∗ a. Σ_(k=1) ^n C_n ^k (((−1)^k )/k) = Σ_(k=1) ^n (1/k) b. Σ_(k=1) ^n C_n ^k (((−1)^k )/(2k+1)) = (4^n /((2n+1)C_(2n) ^n ))

$$\mathrm{Prove}\:\mathrm{that}\::\:\forall\mathrm{n}\in\mathbb{N}^{\ast} \\ $$$$\mathrm{a}.\:\:\:\:\:\:\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}{C}_{\mathrm{n}} ^{\mathrm{k}} \frac{\left(−\mathrm{1}\right)^{\mathrm{k}} }{\mathrm{k}}\:=\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{1}}{\mathrm{k}} \\ $$$$\mathrm{b}.\:\:\:\:\:\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}{C}_{\mathrm{n}} ^{\mathrm{k}} \:\frac{\left(−\mathrm{1}\right)^{\mathrm{k}} }{\mathrm{2k}+\mathrm{1}}\:=\:\frac{\mathrm{4}^{\mathrm{n}} }{\left(\mathrm{2n}+\mathrm{1}\right){C}_{\mathrm{2n}} ^{\mathrm{n}} } \\ $$

Question Number 143474 Answers: 0 Comments: 0

for all positive integral., u_(n+1) =u_n (u_(n−1) ^2 −2)−u_n u_n =2 and u_1 =2(1/2) prove that : 3log_2 [u_n ]=2^n −1(−1)^n where [x] is the integral part of x

$${for}\:{all}\:{positive}\:{integral}., \\ $$$$\:\mathrm{u}_{\mathrm{n}+\mathrm{1}} =\mathrm{u}_{\mathrm{n}} \left(\mathrm{u}_{\mathrm{n}−\mathrm{1}} ^{\mathrm{2}} −\mathrm{2}\right)−\mathrm{u}_{\mathrm{n}} \\ $$$$\:\mathrm{u}_{\mathrm{n}} =\mathrm{2}\:{and}\:\mathrm{u}_{\mathrm{1}} =\mathrm{2}\frac{\mathrm{1}}{\mathrm{2}} \\ $$$${prove}\:{that}\::\:\mathrm{3log}_{\mathrm{2}} \left[\mathrm{u}_{\mathrm{n}} \right]=\mathrm{2}^{\mathrm{n}} −\mathrm{1}\left(−\mathrm{1}\right)^{\mathrm{n}} \\ $$$${where}\:\left[\mathrm{x}\right]\:{is}\:{the}\:{integral}\:{part}\:{of}\:\:\mathrm{x} \\ $$

Question Number 143453 Answers: 1 Comments: 0

Question Number 143454 Answers: 1 Comments: 0

........nice .......integral....... T :=∫_0 ^( ∞) ((arctan(x))/x^( ln(x) +1) )dx=^? ((π(√π))/4)

$$ \\ $$$$\:\:\:........{nice}\:.......{integral}....... \\ $$$$\:\:\mathscr{T}\:\::=\int_{\mathrm{0}} ^{\:\infty} \frac{{arctan}\left({x}\right)}{{x}^{\:{ln}\left({x}\right)\:+\mathrm{1}} }{dx}\overset{?} {=}\frac{\pi\sqrt{\pi}}{\mathrm{4}} \\ $$$$ \\ $$

Question Number 143450 Answers: 2 Comments: 0

when x+y=((2π)/3); x≥0 ;y≥0 the maximum and the minimum of sin x+sin y is ___

$${when}\:{x}+{y}=\frac{\mathrm{2}\pi}{\mathrm{3}};\:{x}\geqslant\mathrm{0}\:;{y}\geqslant\mathrm{0} \\ $$$${the}\:{maximum}\:{and}\:{the}\:{minimum} \\ $$$${of}\:\mathrm{sin}\:{x}+\mathrm{sin}\:{y}\:{is}\:\_\_\_\: \\ $$

Question Number 145524 Answers: 1 Comments: 0

What is the argument of the complex numbers below (i) z = 1+e^((π/6)i) (ii) z = 1 −e^((π/6)i)

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{argument}\:\mathrm{of}\:\mathrm{the}\:\mathrm{complex}\:\mathrm{numbers}\:\mathrm{below} \\ $$$$\left(\mathrm{i}\right)\:{z}\:=\:\mathrm{1}+{e}^{\frac{\pi}{\mathrm{6}}{i}} \\ $$$$\left(\mathrm{ii}\right)\:{z}\:=\:\mathrm{1}\:−{e}^{\frac{\pi}{\mathrm{6}}{i}} \\ $$

Question Number 143446 Answers: 1 Comments: 0

find general solution of differential equation y′=y−xy^3 e^(−2x)

$$\mathrm{find}\:\mathrm{general}\:\mathrm{solution}\:\mathrm{of} \\ $$$$\mathrm{differential}\:\mathrm{equation}\: \\ $$$$\mathrm{y}'=\mathrm{y}−\mathrm{xy}^{\mathrm{3}} \mathrm{e}^{−\mathrm{2x}} \\ $$

Question Number 143443 Answers: 1 Comments: 1

Question Number 143545 Answers: 2 Comments: 0

calculate ∫_0 ^∞ ((logx)/(x^6 +1))dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{logx}}{\mathrm{x}^{\mathrm{6}} \:+\mathrm{1}}\mathrm{dx} \\ $$

Question Number 143439 Answers: 2 Comments: 0

sinx+sin2x+sin3x+.....+sinkx=sin((kx)/2)×((sin((k+1)/2)x)/(sin(x/2))) prove

$$\mathrm{sinx}+\mathrm{sin2x}+\mathrm{sin3x}+.....+\mathrm{sinkx}=\mathrm{sin}\frac{\mathrm{kx}}{\mathrm{2}}×\frac{\mathrm{sin}\frac{\mathrm{k}+\mathrm{1}}{\mathrm{2}}\mathrm{x}}{\mathrm{sin}\frac{\mathrm{x}}{\mathrm{2}}} \\ $$$$\mathrm{prove} \\ $$

Question Number 143438 Answers: 1 Comments: 0

((1/2))^x =log_(1/2) x find x

$$\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\boldsymbol{{x}}} =\boldsymbol{{log}}_{\frac{\mathrm{1}}{\mathrm{2}}} \boldsymbol{{x}} \\ $$$$\boldsymbol{{find}}\:\:\:\boldsymbol{{x}} \\ $$

Question Number 143435 Answers: 0 Comments: 5

Question Number 143436 Answers: 0 Comments: 0

Find matrix rank=? (((47),(−67),(35),(201),(155)),((26),(98),(23),(−294),(86)),((16),(−428),1,(1284),(53)) )

$${Find}\:{matrix}\:{rank}=? \\ $$$$\begin{pmatrix}{\mathrm{47}}&{−\mathrm{67}}&{\mathrm{35}}&{\mathrm{201}}&{\mathrm{155}}\\{\mathrm{26}}&{\mathrm{98}}&{\mathrm{23}}&{−\mathrm{294}}&{\mathrm{86}}\\{\mathrm{16}}&{−\mathrm{428}}&{\mathrm{1}}&{\mathrm{1284}}&{\mathrm{53}}\end{pmatrix} \\ $$

Question Number 143431 Answers: 0 Comments: 0

Question Number 143429 Answers: 1 Comments: 0

....... nice .....integral....... Evaluate :: ξ := ∫_0 ^( 1) ((ln(1−t))/(1+t^2 )) dt =?

$$\:\: \\ $$$$\:\:\:\:\:\:\:.......\:{nice}\:.....{integral}....... \\ $$$$\:\:\:\:{Evaluate}\::: \\ $$$$\:\:\:\:\:\:\:\:\xi\::=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}−{t}\right)}{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt}\:=? \\ $$

Question Number 143428 Answers: 1 Comments: 0

...Advanced ......Mathematics... Evaluate:: 𝛗 :=Σ_(n=1) ^∞ ((coth(πn))/n^3 ) =?

$$\:\:\:\:\: \\ $$$$...{Advanced}\:......{Mathematics}... \\ $$$$\:\:\:\:\:{Evaluate}:: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\phi}\::=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{coth}\left(\pi{n}\right)}{{n}^{\mathrm{3}} }\:=? \\ $$

Question Number 143427 Answers: 1 Comments: 0

if f(x) is polynomial satisfying f(x)f((1/x))−2f(x)+2f((1/x))=5 and f(2)=14 then f(3)=?

$${if}\:{f}\left({x}\right)\:{is}\:{polynomial}\:{satisfying} \\ $$$${f}\left({x}\right){f}\left(\frac{\mathrm{1}}{{x}}\right)−\mathrm{2}{f}\left({x}\right)+\mathrm{2}{f}\left(\frac{\mathrm{1}}{{x}}\right)=\mathrm{5} \\ $$$${and}\:{f}\left(\mathrm{2}\right)=\mathrm{14}\:{then}\:{f}\left(\mathrm{3}\right)=? \\ $$

Question Number 143418 Answers: 1 Comments: 0

Question Number 143414 Answers: 1 Comments: 0

Evaluate ⌊(({Σ_(i=1) ^(1010) tan^2 (((iπ)/(2021)))}^(1/2) )/(Π_(i=1) ^(1010) tan (((iπ)/(2021)))))⌋ wherw⌊∙⌋ denotes GIF

$${Evaluate}\: \\ $$$$\lfloor\frac{\left\{\underset{{i}=\mathrm{1}} {\overset{\mathrm{1010}} {\sum}}\:{tan}^{\mathrm{2}} \left(\frac{{i}\pi}{\mathrm{2021}}\right)\right\}^{\frac{\mathrm{1}}{\mathrm{2}}} }{\underset{{i}=\mathrm{1}} {\overset{\mathrm{1010}} {\prod}}\:{tan}\:\left(\frac{{i}\pi}{\mathrm{2021}}\right)}\rfloor\:\:\:\:{wherw}\lfloor\centerdot\rfloor\:{denotes}\:{GIF} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 143410 Answers: 2 Comments: 4

if x;y;z>0 prove that... ((x/z))^2 e^(((z/x))^2 ) + ((y/x))^2 e^(((x/y))^2 ) + ((z/y))^2 e^(((y/z))^2 ) ≥ 3e

$${if}\:\:{x};{y};{z}>\mathrm{0}\:\:{prove}\:{that}... \\ $$$$\left(\frac{{x}}{{z}}\right)^{\mathrm{2}} \boldsymbol{{e}}^{\left(\frac{{z}}{{x}}\right)^{\mathrm{2}} } +\:\left(\frac{{y}}{{x}}\right)^{\mathrm{2}} \boldsymbol{{e}}^{\left(\frac{{x}}{{y}}\right)^{\mathrm{2}} } +\:\left(\frac{{z}}{{y}}\right)^{\mathrm{2}} \:\boldsymbol{{e}}^{\left(\frac{{y}}{{z}}\right)^{\mathrm{2}} } \geqslant\:\mathrm{3}\boldsymbol{{e}} \\ $$

Question Number 143399 Answers: 2 Comments: 0

∫_0 ^(2π) (dx/((1−ksinx)^2 ))

$$\int_{\mathrm{0}} ^{\mathrm{2}\pi} \frac{\mathrm{dx}}{\left(\mathrm{1}−\mathrm{ksinx}\right)^{\mathrm{2}} } \\ $$

Question Number 143393 Answers: 1 Comments: 0

Given that f(r)=(r+1)! ∙ r, show that f(r)−f(r−1)=r!(r^2 +1). Hence or otherwise, show that 2! ∙ 5+3! ∙ 10+4! ∙ 17+......n!(n^2 +1)=(n+1)! ∙ (n−2)

$$\mathrm{Given}\:\mathrm{that}\:{f}\left({r}\right)=\left({r}+\mathrm{1}\right)!\:\centerdot\:{r},\:\mathrm{show}\:\mathrm{that} \\ $$$${f}\left({r}\right)−{f}\left({r}−\mathrm{1}\right)={r}!\left({r}^{\mathrm{2}} +\mathrm{1}\right). \\ $$$$\mathrm{Hence}\:\mathrm{or}\:\mathrm{otherwise},\:\mathrm{show}\:\mathrm{that} \\ $$$$\mathrm{2}!\:\centerdot\:\mathrm{5}+\mathrm{3}!\:\centerdot\:\mathrm{10}+\mathrm{4}!\:\centerdot\:\mathrm{17}+......{n}!\left({n}^{\mathrm{2}} +\mathrm{1}\right)=\left({n}+\mathrm{1}\right)!\:\centerdot\:\left({n}−\mathrm{2}\right) \\ $$

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