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Question Number 36837    Answers: 1   Comments: 1

Σ_(r=0) ^∞ 2^(4r−2)

$$\underset{{r}=\mathrm{0}} {\overset{\infty} {\sum}}\mathrm{2}^{\mathrm{4}{r}−\mathrm{2}} \: \\ $$

Question Number 36828    Answers: 0   Comments: 7

Question Number 36821    Answers: 2   Comments: 1

>. 2sin((5π)/(12))sin(π/(12)) slove this.?

$$>.\:\mathrm{2}{sin}\frac{\mathrm{5}\pi}{\mathrm{12}}{sin}\frac{\pi}{\mathrm{12}}\:{slove}\:{this}.? \\ $$

Question Number 36820    Answers: 1   Comments: 1

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

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

Question Number 36819    Answers: 0   Comments: 1

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

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

Question Number 36818    Answers: 1   Comments: 1

find f(a) = ∫ (dx/(√(1−ax^2 ))) with a from R .

$${find}\:{f}\left({a}\right)\:=\:\int\:\:\:\:\:\frac{{dx}}{\sqrt{\mathrm{1}−{ax}^{\mathrm{2}} }}\:\:{with}\:{a}\:{from}\:{R}\:. \\ $$

Question Number 36814    Answers: 1   Comments: 0

now the way is clear, also try this one: ∫((cos x)/(sin^2 x (√(sin 2x))))dx

$$\mathrm{now}\:\mathrm{the}\:\mathrm{way}\:\mathrm{is}\:\mathrm{clear},\:\mathrm{also}\:\mathrm{try}\:\mathrm{this}\:\mathrm{one}: \\ $$$$\int\frac{\mathrm{cos}\:{x}}{\mathrm{sin}^{\mathrm{2}} \:{x}\:\sqrt{\mathrm{sin}\:\mathrm{2}{x}}}{dx} \\ $$

Question Number 36811    Answers: 1   Comments: 0

∫ ((sin x)/(cos^2 x. (√(cos 2x)))) dx= ?

$$\int\:\frac{\mathrm{sin}\:{x}}{\mathrm{cos}\:^{\mathrm{2}} {x}.\:\sqrt{\mathrm{cos}\:\mathrm{2}{x}}}\:{dx}=\:? \\ $$

Question Number 36801    Answers: 2   Comments: 0

∫ ((1+x^4 )/((1−x^4 )^(3/2) )) dx = A ∫ A = B Find B ? Assume integration of constant=0.

$$\int\:\frac{\mathrm{1}+{x}^{\mathrm{4}} }{\left(\mathrm{1}−{x}^{\mathrm{4}} \right)^{\frac{\mathrm{3}}{\mathrm{2}}} }\:{dx}\:=\:{A}\: \\ $$$$\int\:\mathrm{A}\:=\:\mathrm{B} \\ $$$$\mathrm{Find}\:\mathrm{B}\:? \\ $$$$\mathrm{Assume}\:\mathrm{integration}\:\mathrm{of}\:\mathrm{constant}=\mathrm{0}. \\ $$

Question Number 36799    Answers: 1   Comments: 1

find ∫_0 ^∞ e^t ln(1+e^(−2t) )dt .

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\:{e}^{{t}} {ln}\left(\mathrm{1}+{e}^{−\mathrm{2}{t}} \right){dt}\:. \\ $$

Question Number 36784    Answers: 0   Comments: 0

Prove that Σ_(r=0) ^n r ((n),(r) )^2 = n (((2n − 1)),(( n − 1)) )

$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\underset{{r}=\mathrm{0}} {\overset{{n}} {\sum}}\:{r}\:\begin{pmatrix}{{n}}\\{{r}}\end{pmatrix}^{\mathrm{2}} \:=\:{n}\:\begin{pmatrix}{\mathrm{2}{n}\:−\:\mathrm{1}}\\{\:\:{n}\:−\:\mathrm{1}}\end{pmatrix} \\ $$

Question Number 36771    Answers: 0   Comments: 3

i have a suggestion...pls request members of the forum to post four to five question so that we get time to solve them...there is flood of questions...so little time to see all post pls give comment if you agree wkth it...

$${i}\:{have}\:{a}\:{suggestion}...{pls}\:{request}\:{members} \\ $$$${of}\:{the}\:{forum}\:{to}\:{post}\:{four}\:{to}\:{five}\:{question} \\ $$$${so}\:{that}\:{we}\:{get}\:{time}\:{to}\:{solve}\:{them}...{there}\:{is}\: \\ $$$${flood}\:{of}\:{questions}...{so}\:{little}\:{time}\:{to}\:{see}\:{all}\:{post} \\ $$$${pls}\:{give}\:{comment}\:{if}\:{you}\:{agree}\:{wkth}\:{it}... \\ $$

Question Number 36762    Answers: 1   Comments: 2

find A_n = ∫_0 ^(π/4) (cosx +sinx)^n dx.

$${find}\:{A}_{{n}} \:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\left({cosx}\:+{sinx}\right)^{{n}} \:{dx}. \\ $$

Question Number 36755    Answers: 1   Comments: 4

let f(a) = ∫_0 ^1 e^t ln(1+ e^(−at) )dt with a≥0 1) find f(a) 2) calculate f^′ (a) 3) find the value of ∫_0 ^1 e^t ln(1+e^(−3t) )dt .

$${let}\:{f}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{e}^{{t}} {ln}\left(\mathrm{1}+\:{e}^{−{at}} \right){dt}\:\:{with}\:{a}\geqslant\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{f}^{'} \left({a}\right) \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{e}^{{t}} {ln}\left(\mathrm{1}+{e}^{−\mathrm{3}{t}} \right){dt}\:. \\ $$

Question Number 36754    Answers: 1   Comments: 1

calculate ∫_1 ^(+∞) (dx/(x^2 (√(4+x^2 )))) .

$${calculate}\:\:\:\int_{\mathrm{1}} ^{+\infty} \:\:\:\:\frac{{dx}}{{x}^{\mathrm{2}} \sqrt{\mathrm{4}+{x}^{\mathrm{2}} }}\:. \\ $$

Question Number 36753    Answers: 1   Comments: 2

find I_n = ∫_0 ^1 x^n arctan(x)dx .

$${find}\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:{x}^{{n}} \:{arctan}\left({x}\right){dx}\:. \\ $$

Question Number 36752    Answers: 1   Comments: 4

find ∫ (dx/(arcsinx(√(1−x^2 )))) .

$${find}\:\:\int\:\:\:\frac{{dx}}{{arcsinx}\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}\:. \\ $$

Question Number 36751    Answers: 0   Comments: 0

let f(x)= Σ_(n=1) ^∞ x^n^2 with x∈]−1,1[ prove that f(x) ∼ ((√π)/(2(√(−ln(x))))) (x →1^− )

$$\left.{let}\:\:{f}\left({x}\right)=\:\sum_{{n}=\mathrm{1}} ^{\infty} \:{x}^{{n}^{\mathrm{2}} } \:\:\:{with}\:\:{x}\in\right]−\mathrm{1},\mathrm{1}\left[\right. \\ $$$${prove}\:{that}\:\:{f}\left({x}\right)\:\sim\:\frac{\sqrt{\pi}}{\mathrm{2}\sqrt{−{ln}\left({x}\right)}}\:\left({x}\:\rightarrow\mathrm{1}^{−} \right) \\ $$

Question Number 36750    Answers: 0   Comments: 0

let f(t)=Σ_(n≥1) (−1)^n ln{1+ (t^2 /(n(1+t^2 )))} 1) study the simple and uniform convergence of Σ f_n 2)study the continuity of f 3) prove that lim_(t→+∞) f(t)=ln((2/π)) .

$${let}\:{f}\left({t}\right)=\sum_{{n}\geqslant\mathrm{1}} \:\left(−\mathrm{1}\right)^{{n}} {ln}\left\{\mathrm{1}+\:\frac{{t}^{\mathrm{2}} }{{n}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}\right\} \\ $$$$\left.\mathrm{1}\right)\:{study}\:{the}\:{simple}\:\:{and}\:{uniform}\:{convergence} \\ $$$${of}\:\Sigma\:{f}_{{n}} \\ $$$$\left.\mathrm{2}\right){study}\:{the}\:{continuity}\:{of}\:{f} \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:{lim}_{{t}\rightarrow+\infty} \:{f}\left({t}\right)={ln}\left(\frac{\mathrm{2}}{\pi}\right)\:. \\ $$

Question Number 36748    Answers: 0   Comments: 0

let f(x)= Σ_(n=1) ^∞ (((−1)^(n−1) )/(ln(nx))) 1) give D_f and study f on]1,+∞[ 2)study the continjity of f and calculate lim _(x→1) f(x) and lim_(x→+∞) f(x). 3) prove that f is C^1 on ]1,+∞[ .

$${let}\:{f}\left({x}\right)=\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} }{{ln}\left({nx}\right)} \\ $$$$\left.\mathrm{1}\left.\right)\:{give}\:{D}_{{f}} \:\:{and}\:{study}\:{f}\:{on}\right]\mathrm{1},+\infty\left[\right. \\ $$$$\left.\mathrm{2}\right){study}\:{the}\:{continjity}\:{of}\:{f}\:{and}\:{calculate} \\ $$$${lim}\:_{{x}\rightarrow\mathrm{1}} {f}\left({x}\right)\:{and}\:{lim}_{{x}\rightarrow+\infty} {f}\left({x}\right). \\ $$$$\left.\mathrm{3}\left.\right)\:{prove}\:{that}\:{f}\:{is}\:{C}^{\mathrm{1}} \:{on}\:\right]\mathrm{1},+\infty\left[\:.\right. \\ $$

Question Number 36747    Answers: 0   Comments: 1

let f(x)= Σ_(n=1) ^∞ ((sin(nx))/n) x^n 1) prove that f is C^1 on ]−1,1[ 2)calculate f^′ (x) and prove that f(x)=arctan( ((xsinx)/(1−x cosx)))

$${let}\:{f}\left({x}\right)=\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{{sin}\left({nx}\right)}{{n}}\:{x}^{{n}} \\ $$$$\left.\mathrm{1}\left.\right)\:{prove}\:{that}\:{f}\:{is}\:{C}^{\mathrm{1}} \:{on}\:\right]−\mathrm{1},\mathrm{1}\left[\right. \\ $$$$\left.\mathrm{2}\right){calculate}\:{f}^{'} \left({x}\right)\:{and}\:{prove}\:{that} \\ $$$${f}\left({x}\right)={arctan}\left(\:\frac{{xsinx}}{\mathrm{1}−{x}\:{cosx}}\right) \\ $$

Question Number 36746    Answers: 0   Comments: 0

solve the d.e. y^(′′) (t) +y(t)=Σ_(n=0) ^N a_n cos(nt)

$${solve}\:{the}\:{d}.{e}.\:{y}^{''} \left({t}\right)\:+{y}\left({t}\right)=\sum_{{n}=\mathrm{0}} ^{{N}} \:{a}_{{n}} {cos}\left({nt}\right) \\ $$

Question Number 36745    Answers: 0   Comments: 0

solve the d.e y^(′′) (t) +y(t)=cos(nt)

$${solve}\:{the}\:{d}.{e}\:\:{y}^{''} \left({t}\right)\:+{y}\left({t}\right)={cos}\left({nt}\right) \\ $$

Question Number 36744    Answers: 0   Comments: 0

let f(x)=Σ_(n=1) ^∞ (1/n) cos^n (x)sin(nx) 1)prove the convergence of this serie 2)prove that f is C^2 on R −{kπ,k∈Z}and calculate f^′ (x) 3) give a exprrssion of f.

$${let}\:{f}\left({x}\right)=\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\mathrm{1}}{{n}}\:{cos}^{{n}} \left({x}\right){sin}\left({nx}\right) \\ $$$$\left.\mathrm{1}\right){prove}\:{the}\:{convergence}\:{of}\:{this}\:{serie} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:{f}\:{is}\:{C}^{\mathrm{2}} \:{on}\:{R}\:−\left\{{k}\pi,{k}\in{Z}\right\}{and} \\ $$$${calculate}\:{f}^{'} \left({x}\right) \\ $$$$\left.\mathrm{3}\right)\:{give}\:{a}\:{exprrssion}\:{of}\:{f}. \\ $$

Question Number 36742    Answers: 0   Comments: 1

study the convergence of Σ_(n=1) ^∞ (−1)^n ln(1+ (1/(n(1+x)))).

$${study}\:{the}\:{convergence}\:{of}\: \\ $$$$\sum_{{n}=\mathrm{1}} ^{\infty} \left(−\mathrm{1}\right)^{{n}} {ln}\left(\mathrm{1}+\:\frac{\mathrm{1}}{{n}\left(\mathrm{1}+{x}\right)}\right). \\ $$

Question Number 36741    Answers: 1   Comments: 1

calculate S(x)=Σ_(n=0) ^∞ ((sin(nx))/(n!))

$${calculate}\:{S}\left({x}\right)=\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{{sin}\left({nx}\right)}{{n}!} \\ $$

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