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Question Number 59175 Answers: 0 Comments: 1

find the value of ∫_0 ^∞ ((1−cos(x))/x^2 )dx

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

Question Number 59174 Answers: 0 Comments: 0

calculate ∫∫_([0,2]^2 ) (x+1−(√y))(y+1−(√x))dxdy

$${calculate}\:\int\int_{\left[\mathrm{0},\mathrm{2}\right]^{\mathrm{2}} } \:\:\:\:\left({x}+\mathrm{1}−\sqrt{{y}}\right)\left({y}+\mathrm{1}−\sqrt{{x}}\right){dxdy}\: \\ $$

Question Number 59172 Answers: 0 Comments: 1

calculate ∫∫_([1,3]^2 ) (x+y)ln(x^2 +y^2 )dxdy

$${calculate}\:\int\int_{\left[\mathrm{1},\mathrm{3}\right]^{\mathrm{2}} } \:\:\:\:\left({x}+{y}\right){ln}\left({x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \right){dxdy}\: \\ $$

Question Number 59171 Answers: 0 Comments: 0

let f(x,y) ((arctan(x+2y))/(x +y^2 )) calculate (∂f/∂x)(x,y) , (∂f/∂y)(x,y),(∂^2 f/∂x^2 )(x,y), (∂^2 f/∂y^2 )(x,y) , (∂^2 f/(∂x∂y))(x,y) (∂^2 f/(∂y∂x))(x,y)

$${let}\:{f}\left({x},{y}\right)\:\:\frac{{arctan}\left({x}+\mathrm{2}{y}\right)}{{x}\:+{y}^{\mathrm{2}} } \\ $$$${calculate}\:\frac{\partial{f}}{\partial{x}}\left({x},{y}\right)\:\:,\:\frac{\partial{f}}{\partial{y}}\left({x},{y}\right),\frac{\partial^{\mathrm{2}} {f}}{\partial{x}^{\mathrm{2}} }\left({x},{y}\right),\:\frac{\partial^{\mathrm{2}} {f}}{\partial{y}^{\mathrm{2}} }\left({x},{y}\right)\:,\:\frac{\partial^{\mathrm{2}} {f}}{\partial{x}\partial{y}}\left({x},{y}\right) \\ $$$$\frac{\partial^{\mathrm{2}} {f}}{\partial{y}\partial{x}}\left({x},{y}\right) \\ $$

Question Number 59170 Answers: 1 Comments: 0

Iff(x)= determinant (((sec x),(cos x),(sec^2 x+cosec x cot x)),((cos^2 x),(cos^2 x),( cosec^2 x)),(( 1),(cos^2 x),( cos^2 x))) then ∫_( 0) ^(π/2) f(x) dx =

$$\mathrm{If}{f}\left({x}\right)=\begin{vmatrix}{\mathrm{sec}\:{x}}&{\mathrm{cos}\:{x}}&{\mathrm{sec}^{\mathrm{2}} {x}+\mathrm{cosec}\:{x}\:\mathrm{cot}\:{x}}\\{\mathrm{cos}^{\mathrm{2}} {x}}&{\mathrm{cos}^{\mathrm{2}} {x}}&{\:\:\:\:\:\:\:\:\:\:\mathrm{cosec}^{\mathrm{2}} {x}}\\{\:\:\:\mathrm{1}}&{\mathrm{cos}^{\mathrm{2}} {x}}&{\:\:\:\:\:\:\:\:\:\:\mathrm{cos}^{\mathrm{2}} {x}}\end{vmatrix} \\ $$$$\mathrm{then}\:\underset{\:\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\:{f}\left({x}\right)\:{dx}\:= \\ $$

Question Number 59169 Answers: 0 Comments: 1

calculate A_n =∫∫_([(1/n),n[^2 ) e^(−x^2 −3y^2 ) (√(x^2 +3y^2 ))dxdy and find lim_(n→+∞) A_n

$${calculate}\:{A}_{{n}} =\int\int_{\left[\frac{\mathrm{1}}{{n}},{n}\left[^{\mathrm{2}} \right.\right.} \:\:\:\:{e}^{−{x}^{\mathrm{2}} −\mathrm{3}{y}^{\mathrm{2}} } \sqrt{{x}^{\mathrm{2}} \:+\mathrm{3}{y}^{\mathrm{2}} }{dxdy} \\ $$$${and}\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} \\ $$

Question Number 59168 Answers: 1 Comments: 0

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

$${calculate}\:\int_{−\mathrm{1}} ^{\mathrm{1}} \:\:\frac{{x}^{\mathrm{2}} \:+\mathrm{3}}{\sqrt{\mathrm{1}+{x}}\:+\sqrt{\mathrm{1}−{x}}}\:{dx} \\ $$

Question Number 59167 Answers: 0 Comments: 0

find the value of Σ_(n=1) ^∞ ((n^2 +1)/(n^3 (n+1)^2 ))

$${find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{n}^{\mathrm{2}} \:+\mathrm{1}}{{n}^{\mathrm{3}} \left({n}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 59166 Answers: 0 Comments: 0

67{[4×(3+5)+6]}

$$\mathrm{67}\left\{\left[\mathrm{4}×\left(\mathrm{3}+\mathrm{5}\right)+\mathrm{6}\right]\right\} \\ $$

Question Number 59165 Answers: 0 Comments: 0

let D ={ (x,y)∈R^2 / x>0 ,y>0 and x+y ≤2 } calculate ∫∫_D (x+y −(√(x+y)))dxdy

$${let}\:{D}\:=\left\{\:\left({x},{y}\right)\in{R}^{\mathrm{2}} /\:{x}>\mathrm{0}\:,{y}>\mathrm{0}\:\:{and}\:\:{x}+{y}\:\leqslant\mathrm{2}\:\right\} \\ $$$${calculate}\:\int\int_{{D}} \:\left({x}+{y}\:−\sqrt{{x}+{y}}\right){dxdy}\: \\ $$

Question Number 59164 Answers: 1 Comments: 0

5^x =0 find x

$$\mathrm{5}^{\mathrm{x}} =\mathrm{0} \\ $$$$\mathrm{find}\:\mathrm{x} \\ $$

Question Number 59163 Answers: 0 Comments: 0

calculate A_n =∫∫_W_n ((1−(√(x^2 +y^2 )))/(1+(√(x^2 +y^2 )))) dxdy with W_n =](1/n),n[^2 2) find lim_(n→+∞) A_n

$$\left.{calculate}\:{A}_{{n}} =\int\int_{{W}_{{n}} } \:\:\:\frac{\mathrm{1}−\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }}{\mathrm{1}+\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }}\:{dxdy}\:\:\:{with}\:{W}_{{n}} \:=\right]\frac{\mathrm{1}}{{n}},{n}\left[^{\mathrm{2}} \right. \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} \\ $$

Question Number 59162 Answers: 0 Comments: 0

calculate ∫∫_D (√(x^2 −y^2 ))xy dxdy with D ={(x,y)∈ R^2 /0≤y≤1 and 2 ≤x ≤5 }

$${calculate}\:\:\int\int_{{D}} \:\:\sqrt{{x}^{\mathrm{2}} −{y}^{\mathrm{2}} }{xy}\:{dxdy}\:\:{with} \\ $$$${D}\:=\left\{\left({x},{y}\right)\in\:{R}^{\mathrm{2}} \:\:\:/\mathrm{0}\leqslant{y}\leqslant\mathrm{1}\:\:{and}\:\:\mathrm{2}\:\leqslant{x}\:\leqslant\mathrm{5}\:\right\} \\ $$

Question Number 59161 Answers: 0 Comments: 0

calculatef(a)= ∫_0 ^∞ ((ln(a^2 +x^2 ))/(a^2 +x^2 ))dx with >0 1) calculate ∫_0 ^∞ ((ln(2+x^2 ))/(2+x^2 ))dx and ∫_0 ^∞ ((ln(3+x^2 ))/(3+x^2 )) dx .

$${calculatef}\left({a}\right)=\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{ln}\left({a}^{\mathrm{2}} +{x}^{\mathrm{2}} \right)}{{a}^{\mathrm{2}} +{x}^{\mathrm{2}} }{dx}\:\:\:{with}\:>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{ln}\left(\mathrm{2}+{x}^{\mathrm{2}} \right)}{\mathrm{2}+{x}^{\mathrm{2}} }{dx}\:\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{ln}\left(\mathrm{3}+{x}^{\mathrm{2}} \right)}{\mathrm{3}+{x}^{\mathrm{2}} }\:{dx}\:. \\ $$

Question Number 59152 Answers: 0 Comments: 1

Question Number 59147 Answers: 2 Comments: 0

Σ_(1°) ^(89°) log_2 tan r°

$$\underset{\mathrm{1}°} {\overset{\mathrm{89}°} {\sum}}\:{log}_{\mathrm{2}} {tan}\:{r}° \\ $$

Question Number 59135 Answers: 0 Comments: 0

Question Number 59133 Answers: 2 Comments: 0

The total number of terms in the expression (x + a)^(100) + (x − a)^(100) after simplification is ?

$$\mathrm{The}\:\mathrm{total}\:\mathrm{number}\:\mathrm{of}\:\mathrm{terms}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expression}\:\:\:\left(\mathrm{x}\:+\:\mathrm{a}\right)^{\mathrm{100}} \:+\:\left(\mathrm{x}\:−\:\mathrm{a}\right)^{\mathrm{100}} \\ $$$$\mathrm{after}\:\mathrm{simplification}\:\mathrm{is}\:\:? \\ $$

Question Number 59131 Answers: 2 Comments: 0

a, b, c ∈ R a + b + c = 5 Prove that (√(a^2 + b^2 − 2b + 1)) + (√(b^2 + c^2 − 2c + 1)) + (√(c^2 + a^2 − 2a + 1)) ≥ (√(29))

$${a},\:{b},\:{c}\:\:\in\:\:\mathbb{R} \\ $$$${a}\:+\:{b}\:+\:{c}\:\:=\:\:\mathrm{5} \\ $$$${Prove}\:\:{that} \\ $$$$\sqrt{{a}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} \:−\:\mathrm{2}{b}\:+\:\mathrm{1}}\:\:+\:\:\sqrt{{b}^{\mathrm{2}} \:+\:{c}^{\mathrm{2}} \:−\:\mathrm{2}{c}\:+\:\mathrm{1}}\:\:+\:\:\sqrt{{c}^{\mathrm{2}} \:+\:{a}^{\mathrm{2}} \:−\:\mathrm{2}{a}\:+\:\mathrm{1}}\:\:\:\geqslant\:\:\sqrt{\mathrm{29}} \\ $$

Question Number 59129 Answers: 0 Comments: 0

Question Number 59121 Answers: 1 Comments: 0

Question Number 59114 Answers: 4 Comments: 1

Question Number 59113 Answers: 1 Comments: 0

Let a is a real number . How many solutions can the equation in θ (sin θ + cos θ)(sin θ cos θ − 1) = a have for 0 < θ < (π/2) ?

$${Let}\:\:{a}\:\:{is}\:\:{a}\:\:{real}\:\:{number}\:.\:\:{How}\:\:{many}\:\:{solutions} \\ $$$${can}\:\:{the}\:\:{equation}\:\:{in}\:\:\theta\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{sin}\:\theta\:+\:\mathrm{cos}\:\theta\right)\left(\mathrm{sin}\:\theta\:\mathrm{cos}\:\theta\:−\:\mathrm{1}\right)\:\:=\:\:{a} \\ $$$${have}\:\:{for}\:\:\mathrm{0}\:<\:\theta\:<\:\frac{\pi}{\mathrm{2}}\:\:? \\ $$

Question Number 59104 Answers: 0 Comments: 0

i want of ask in mechanics is this possible?? power=((work done)/(time taken)) if power = p_w , Work= W_d , and time=t ⇒ p_w = (W_d /t) but W_d = force(f)×distance(s) W_d = fs ⇒ p_w = ((f×s)/t) p_w = f × (s/t)(distance/time) p_w = f × velocity but f=ma p_w = m×a×v rearranging ⇒ p_(w ) = m×v×a p_w = momentum(P)×Acceleration(a) Power = momentum × Acceleration. the momentum of a system is directly propotional to the power of that system but will have a minimum momentum when accelerating. what do you think?

$${i}\:{want}\:{of}\:{ask}\:{in}\:{mechanics}\:{is}\:{this} \\ $$$${possible}?? \\ $$$$ \\ $$$${power}=\frac{{work}\:{done}}{{time}\:{taken}} \\ $$$${if}\:{power}\:=\:{p}_{{w}} \:,\:{Work}=\:{W}_{{d}} ,\:{and}\:{time}={t} \\ $$$$\Rightarrow\:{p}_{{w}} =\:\frac{{W}_{{d}} }{{t}} \\ $$$${but}\:{W}_{{d}} =\:{force}\left({f}\right)×{distance}\left({s}\right) \\ $$$$\:\:\:\:\:\:\:\:{W}_{{d}} =\:{fs} \\ $$$$\Rightarrow\:{p}_{{w}} =\:\frac{{f}×{s}}{{t}} \\ $$$$\:\:\:\:\:{p}_{{w}} =\:{f}\:×\:\frac{{s}}{{t}}\left({distance}/{time}\right) \\ $$$$\:\:\:{p}_{{w}} =\:{f}\:×\:{velocity} \\ $$$$\:\:\:{but}\:{f}={ma} \\ $$$${p}_{{w}} =\:{m}×{a}×{v} \\ $$$${rearranging} \\ $$$$\Rightarrow\:{p}_{{w}\:} =\:{m}×{v}×{a} \\ $$$$\:\:\:\:\:\:{p}_{{w}} =\:{momentum}\left({P}\right)×{Acceleration}\left({a}\right) \\ $$$${Power}\:=\:{momentum}\:×\:{Acceleration}. \\ $$$${the}\:{momentum}\:{of}\:{a}\:{system}\:{is}\:{directly} \\ $$$${propotional}\:{to}\:{the}\:{power}\:{of}\:{that}\:{system} \\ $$$${but}\:{will}\:{have}\:{a}\:{minimum}\:{momentum} \\ $$$${when}\:{accelerating}. \\ $$$$ \\ $$$${what}\:{do}\:{you}\:{think}? \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 59103 Answers: 0 Comments: 0

Question Number 59102 Answers: 1 Comments: 0

use remainder theorem to factorize completetly the expression x^3 (y − z) + y^3 (z − x) + z^3 (x − y)

$$\mathrm{use}\:\mathrm{remainder}\:\mathrm{theorem}\:\mathrm{to}\:\mathrm{factorize}\:\mathrm{completetly}\:\mathrm{the}\:\mathrm{expression} \\ $$$$\:\:\:\:\:\mathrm{x}^{\mathrm{3}} \left(\mathrm{y}\:−\:\mathrm{z}\right)\:+\:\mathrm{y}^{\mathrm{3}} \left(\mathrm{z}\:−\:\mathrm{x}\right)\:+\:\mathrm{z}^{\mathrm{3}} \left(\mathrm{x}\:−\:\mathrm{y}\right) \\ $$

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