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Question Number 26824    Answers: 3   Comments: 0

Question Number 26819    Answers: 0   Comments: 0

Question Number 26812    Answers: 1   Comments: 0

sum of infinite seris tan^(−1) (2/n^2 )

$${sum}\:{of}\:{infinite}\:{seris} \\ $$$$\mathrm{tan}^{−\mathrm{1}} \left(\mathrm{2}/{n}^{\mathrm{2}} \right) \\ $$

Question Number 26799    Answers: 1   Comments: 0

Question Number 26801    Answers: 0   Comments: 0

find expansion of α^3 +β^3 +γ^3

$${find}\:{expansion}\:{of}\:\alpha^{\mathrm{3}} +\beta^{\mathrm{3}} +\gamma^{\mathrm{3}} \\ $$

Question Number 26781    Answers: 1   Comments: 4

Question Number 26778    Answers: 1   Comments: 1

If sin(A) + sin(B) = p and cos(A) + cos(B) = q show that tan(A) + tan(B) = ((8pq)/((p^2 + q^2 )^2 − 4q^2 ))

$$\mathrm{If}\:\:\:\:\:\:\:\mathrm{sin}\left(\mathrm{A}\right)\:+\:\mathrm{sin}\left(\mathrm{B}\right)\:=\:\mathrm{p}\:\:\:\:\:\mathrm{and}\:\:\:\:\mathrm{cos}\left(\mathrm{A}\right)\:+\:\mathrm{cos}\left(\mathrm{B}\right)\:=\:\mathrm{q} \\ $$$$\mathrm{show}\:\mathrm{that}\:\:\:\:\mathrm{tan}\left(\mathrm{A}\right)\:+\:\mathrm{tan}\left(\mathrm{B}\right)\:=\:\frac{\mathrm{8pq}}{\left(\mathrm{p}^{\mathrm{2}} \:+\:\mathrm{q}^{\mathrm{2}} \right)^{\mathrm{2}} \:−\:\mathrm{4q}^{\mathrm{2}} } \\ $$

Question Number 26776    Answers: 0   Comments: 2

Question Number 26774    Answers: 0   Comments: 5

If one side of a square reduces by 10% and the adjacent side increases by 30%.Using the application of differentiationfind the % change in the area of the square.

$${If}\:{one}\:{side}\:{of}\:{a}\:{square}\:{reduces}\:{by} \\ $$$$\mathrm{10\%}\:{and}\:{the}\:{adjacent}\:{side}\:{increases} \\ $$$${by}\:\mathrm{30\%}.{Using}\:{the}\:{application}\:{of} \\ $$$${differentiationfind}\:{the}\:\%\:{change} \\ $$$${in}\:{the}\:{area}\:{of}\:{the}\:{square}. \\ $$

Question Number 26772    Answers: 1   Comments: 0

With a center on a given circle of radius r ,an arc has been drawn in order to divide the circle in two equal (in area) parts. What is the radius of the arc in terms of r (radius of given circle)?

$$\mathrm{With}\:\mathrm{a}\:\mathrm{center}\:\boldsymbol{\mathrm{on}}\:\mathrm{a}\:\mathrm{given}\:\mathrm{circle}\:\mathrm{of} \\ $$$$\mathrm{radius}\:\mathrm{r}\:,\mathrm{an}\:\mathrm{arc}\:\mathrm{has}\:\mathrm{been}\:\mathrm{drawn}\:\mathrm{in}\:\mathrm{order} \\ $$$$\mathrm{to}\:\mathrm{divide}\:\mathrm{the}\:\mathrm{circle}\:\mathrm{in}\:\mathrm{two}\:\mathrm{equal} \\ $$$$\left(\mathrm{in}\:\mathrm{area}\right)\:\mathrm{parts}. \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{radius}\:\mathrm{of}\:\mathrm{the}\:\mathrm{arc}\:\mathrm{in}\:\mathrm{terms} \\ $$$$\mathrm{of}\:\:\mathrm{r}\:\left(\mathrm{radius}\:\mathrm{of}\:\mathrm{given}\:\mathrm{circle}\right)?\: \\ $$

Question Number 26771    Answers: 0   Comments: 1

∫_( 0) ^( [x]) (2^x /2^([x]) ) dx =

$$\underset{\:\mathrm{0}} {\overset{\:\:\:\left[{x}\right]} {\int}}\frac{\mathrm{2}^{{x}} }{\mathrm{2}^{\left[{x}\right]} }\:{dx}\:= \\ $$

Question Number 26770    Answers: 0   Comments: 1

lim_(n→∞) (((n !)^(1/n) )/n) =

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\left({n}\:!\right)^{\mathrm{1}/{n}} }{{n}}\:= \\ $$

Question Number 26768    Answers: 0   Comments: 0

find the nature of U_n =Σ_(p=0) ^(p=n) (1/C_n ^p ) .

$${find}\:{the}\:{nature}\:{of}\:{U}_{{n}} \:\:=\sum_{{p}=\mathrm{0}} ^{{p}={n}} \:\:\frac{\mathrm{1}}{{C}_{{n}} ^{{p}} }\:\:. \\ $$

Question Number 26769    Answers: 1   Comments: 1

∫_( 0) ^∞ (dx/([x+(√(x^2 +1)) ]^3 )) =

$$\underset{\:\mathrm{0}} {\overset{\infty} {\int}}\:\:\frac{{dx}}{\left[{x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}\:\right]^{\mathrm{3}} }\:=\: \\ $$

Question Number 26764    Answers: 0   Comments: 3

find lim_(n−>∝) (1/n) ln( Π_(k=1) ^(n−1) (1− (k/n))).

$${find}\:\:{lim}_{{n}−>\propto} \:\frac{\mathrm{1}}{{n}}\:{ln}\left(\:\prod_{{k}=\mathrm{1}} ^{{n}−\mathrm{1}} \:\left(\mathrm{1}−\:\frac{{k}}{{n}}\right)\right). \\ $$

Question Number 26761    Answers: 1   Comments: 0

let put s_n = ((Σ_(k=0) ^(k=n) (2k+1))/(Σ_(k=1) ^(k=n) k)) find lim_(n−>∝) s_n

$${let}\:{put}\:\:{s}_{{n}} =\:\frac{\sum_{{k}=\mathrm{0}} ^{{k}={n}} \left(\mathrm{2}{k}+\mathrm{1}\right)}{\sum_{{k}=\mathrm{1}} ^{{k}={n}} \:{k}} \\ $$$${find}\:{lim}_{{n}−>\propto} {s}_{{n}} \\ $$

Question Number 26759    Answers: 1   Comments: 1

find the value of ∫_0 ^( ∝) (dx/((x+1)(x+2)(x+3))) .

$${find}\:\:{the}\:{value}\:{of}\:\:\:\int_{\mathrm{0}} ^{\:\propto} \:\:\frac{{dx}}{\left({x}+\mathrm{1}\right)\left({x}+\mathrm{2}\right)\left({x}+\mathrm{3}\right)}\:. \\ $$

Question Number 26758    Answers: 0   Comments: 1

let give D={( x,y )∈R^2 /x^2 −x +y^2 ≤ 4 and 0≤y≤1} calculate ∫∫_D ln(xy)(√( x^2 +y^2 dxdy ))

$${let}\:{give}\:{D}=\left\{\left(\:\:{x},{y}\:\right)\in\mathbb{R}^{\mathrm{2}} /{x}^{\mathrm{2}} −{x}\:+{y}^{\mathrm{2}} \leqslant\:\mathrm{4}\:{and}\:\:\mathrm{0}\leqslant{y}\leqslant\mathrm{1}\right\} \\ $$$${calculate}\:\int\int_{{D}} {ln}\left({xy}\right)\sqrt{\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} {dxdy}\:\:} \\ $$

Question Number 26757    Answers: 0   Comments: 3

give the decomposition of F(x) = (1/(x^(2n) +1)) inside C[x] then find the value of ∫_0 ^∞ (dx/(1+x^(2n) )) n∈N and n≠o

$${give}\:{the}\:{decomposition}\:{of}\:{F}\left({x}\right)\:=\:\:\:\frac{\mathrm{1}}{{x}^{\mathrm{2}{n}} +\mathrm{1}}\:\:{inside}\:\mathbb{C}\left[{x}\right] \\ $$$${then}\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{2}{n}} }\:\:\:\:\:\:{n}\in\mathbb{N}\:\:{and}\:{n}\neq{o} \\ $$

Question Number 26756    Answers: 0   Comments: 2

prove that ∫_0 ^1 (dx/(x+ e^x )) = Σ_(n=0) ^∝ (((−1)^n )/((n+1)^(n+1) )) A_n with A_n = ∫_0 ^(n+1) t^n e^(−t) dt .

$${prove}\:{that}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dx}}{{x}+\:{e}^{{x}} }\:=\:\sum_{{n}=\mathrm{0}} ^{\propto} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\left({n}+\mathrm{1}\right)^{{n}+\mathrm{1}} }\:{A}_{{n}} \\ $$$${with}\:\:{A}_{{n}} \:=\:\int_{\mathrm{0}} ^{{n}+\mathrm{1}} \:{t}^{{n}} \:{e}^{−{t}} {dt}\:. \\ $$

Question Number 26755    Answers: 1   Comments: 0

find ∫ (dx/(x^6 −1)) .

$${find}\:\:\int\:\:\frac{{dx}}{{x}^{\mathrm{6}} −\mathrm{1}}\:\:. \\ $$

Question Number 26751    Answers: 0   Comments: 1

by using fourier serie find the value of Σ_(n=0) ^∝ (1/((2n+1)^2 ))

$${by}\:{using}\:{fourier}\:{serie}\:{find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{0}} ^{\propto} \:\frac{\mathrm{1}}{\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 26749    Answers: 0   Comments: 1

let give S_n = Σ_(1≤i<j≤n) (1/(i^2 j^2 )) find lim_(n−>∝) S_n .

$${let}\:{give}\:\:{S}_{{n}} \:=\:\:\sum_{\mathrm{1}\leqslant{i}<{j}\leqslant{n}} \:\:\:\frac{\mathrm{1}}{{i}^{\mathrm{2}} {j}^{\mathrm{2}} }\:\:\:{find}\:{lim}_{{n}−>\propto} \:\:{S}_{{n}} \:\:. \\ $$

Question Number 26738    Answers: 1   Comments: 1

Question Number 26733    Answers: 1   Comments: 1

Question Number 26732    Answers: 0   Comments: 1

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