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AllQuestion and Answers: Page 1013

Question Number 117460 Answers: 0 Comments: 2

((2(√2))/(9801))Σ_(n=1) ^∞ (((4n)!(1103+26390n))/((n!)^4 396^(4n) ))=(1/π) (Prove that)

$$\frac{\mathrm{2}\sqrt{\mathrm{2}}}{\mathrm{9801}}\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(\mathrm{4}{n}\right)!\left(\mathrm{1103}+\mathrm{26390}{n}\right)}{\left({n}!\right)^{\mathrm{4}} \mathrm{396}^{\mathrm{4}{n}} }=\frac{\mathrm{1}}{\pi}\:\:\:\left({Prove}\:{that}\right) \\ $$

Question Number 117458 Answers: 1 Comments: 0

(4/3).((16)/(15)).((36)/(35)).((64)/(63)).((100)/(99)).((144)/(143)).((196)/(195)).((256)/(255)).((324)/(323))......∞

$$\frac{\mathrm{4}}{\mathrm{3}}.\frac{\mathrm{16}}{\mathrm{15}}.\frac{\mathrm{36}}{\mathrm{35}}.\frac{\mathrm{64}}{\mathrm{63}}.\frac{\mathrm{100}}{\mathrm{99}}.\frac{\mathrm{144}}{\mathrm{143}}.\frac{\mathrm{196}}{\mathrm{195}}.\frac{\mathrm{256}}{\mathrm{255}}.\frac{\mathrm{324}}{\mathrm{323}}......\infty \\ $$

Question Number 117446 Answers: 2 Comments: 0

Evaluate ∫((3x^2 −5)/(x^4 +6x^2 +25))dx

$$\mathrm{Evaluate}\:\int\frac{\mathrm{3}{x}^{\mathrm{2}} −\mathrm{5}}{{x}^{\mathrm{4}} +\mathrm{6}{x}^{\mathrm{2}} +\mathrm{25}}\mathrm{d}{x} \\ $$

Question Number 117439 Answers: 1 Comments: 0

Question Number 117438 Answers: 4 Comments: 0

∫_(−∞) ^( ∞) cos((1/2)πx^2 )dx

$$ \\ $$$$\:\:\:\:\:\int_{−\infty} ^{\:\infty} \mathrm{cos}\left(\frac{\mathrm{1}}{\mathrm{2}}\pi{x}^{\mathrm{2}} \right){dx}\:\:\:\:\: \\ $$$$ \\ $$

Question Number 117437 Answers: 2 Comments: 0

∫_0 ^∞ (dx/(a^3 +x^3 )) generaly ∫_0 ^∞ (dx/(p+x^n ))

$$\int_{\mathrm{0}} ^{\infty} \frac{{dx}}{{a}^{\mathrm{3}} +{x}^{\mathrm{3}} } \\ $$$$ \\ $$$${generaly} \\ $$$$\int_{\mathrm{0}} ^{\infty} \frac{{dx}}{{p}+{x}^{{n}} } \\ $$

Question Number 117508 Answers: 1 Comments: 0

f(x+1)+f(x−1) = x^2 find f^(−1) (x) =?

$$\mathrm{f}\left(\mathrm{x}+\mathrm{1}\right)+\mathrm{f}\left(\mathrm{x}−\mathrm{1}\right)\:=\:\mathrm{x}^{\mathrm{2}} \: \\ $$$$\mathrm{find}\:\mathrm{f}^{−\mathrm{1}} \left(\mathrm{x}\right)\:=? \\ $$

Question Number 117503 Answers: 1 Comments: 0

Question Number 117434 Answers: 0 Comments: 0

Question Number 117433 Answers: 0 Comments: 0

Question Number 117431 Answers: 0 Comments: 0

Question Number 117422 Answers: 1 Comments: 0

lim_(x→0) ((3tan 4x−4tan 3x)/(3sin 4x−4sin 3x)) =?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{3tan}\:\mathrm{4x}−\mathrm{4tan}\:\mathrm{3x}}{\mathrm{3sin}\:\mathrm{4x}−\mathrm{4sin}\:\mathrm{3x}}\:=? \\ $$

Question Number 117417 Answers: 2 Comments: 0

lim_(x→0^+ ) (tanh ((1/x))−(1/(cosh ((1/x)))))^(1/x) =?

$$\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\left(\mathrm{tanh}\:\left(\frac{\mathrm{1}}{\mathrm{x}}\right)−\frac{\mathrm{1}}{\mathrm{cosh}\:\left(\frac{\mathrm{1}}{\mathrm{x}}\right)}\right)^{\frac{\mathrm{1}}{\mathrm{x}}} =? \\ $$

Question Number 117416 Answers: 3 Comments: 0

lim_(x→∞) ((e^x^2 −cosx)/(sin^2 x))

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{e}^{\mathrm{x}^{\mathrm{2}} } −\mathrm{cosx}}{\mathrm{sin}^{\mathrm{2}} \mathrm{x}} \\ $$

Question Number 117414 Answers: 0 Comments: 0

Question Number 117412 Answers: 3 Comments: 2

lim_(x→(π/6)) ((1−2sin x)/( 1−(√3) tan x)) = ?

$$\underset{{x}\rightarrow\frac{\pi}{\mathrm{6}}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{2sin}\:\mathrm{x}}{\:\mathrm{1}−\sqrt{\mathrm{3}}\:\mathrm{tan}\:\mathrm{x}}\:=\:? \\ $$

Question Number 117409 Answers: 1 Comments: 0

Question Number 117403 Answers: 1 Comments: 1

∫_0 ^1 (arc tan x)^2 dx =?

$$\:\:\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\left(\mathrm{arc}\:\mathrm{tan}\:\mathrm{x}\right)^{\mathrm{2}} \:\mathrm{dx}\:=? \\ $$

Question Number 117396 Answers: 3 Comments: 0

...differential equation... solve : (dy/dx)=(1/(xy+2x^2 y)) general solution =??? m.n.1970

$$\:\:\:\:\:\:\:\:...{differential}\:\:{equation}...\: \\ $$$$ \\ $$$$\:\:\:\:{solve}\:: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\frac{{dy}}{{dx}}=\frac{\mathrm{1}}{{xy}+\mathrm{2}{x}^{\mathrm{2}} {y}} \\ $$$$\:\:\:\:\:\:\:\:\:{general}\:\:{solution}\:=??? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{m}.{n}.\mathrm{1970} \\ $$$$\: \\ $$

Question Number 117398 Answers: 0 Comments: 0

Question Number 117392 Answers: 2 Comments: 0

If log_4 x+(log_4 x)^2 +(log_4 x)^3 +(log_4 x)^4 +...=1 find the value of x.

$$\mathrm{If}\:\:\:\mathrm{log}_{\mathrm{4}} {x}+\left(\mathrm{log}_{\mathrm{4}} {x}\right)^{\mathrm{2}} +\left(\mathrm{log}_{\mathrm{4}} {x}\right)^{\mathrm{3}} +\left(\mathrm{log}_{\mathrm{4}} {x}\right)^{\mathrm{4}} +...=\mathrm{1} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{x}. \\ $$

Question Number 117391 Answers: 1 Comments: 0

(√(2/3))−(√(2/(27)))+(√(2/(75)))−(√(2/(147)))+(√(2/(243)))−(√(2/(363)))+(√(2/(507)))−(√(2/(675)))+(√(2/(867)))−._ ...

$$\sqrt{\frac{\mathrm{2}}{\mathrm{3}}}−\sqrt{\frac{\mathrm{2}}{\mathrm{27}}}+\sqrt{\frac{\mathrm{2}}{\mathrm{75}}}−\sqrt{\frac{\mathrm{2}}{\mathrm{147}}}+\sqrt{\frac{\mathrm{2}}{\mathrm{243}}}−\sqrt{\frac{\mathrm{2}}{\mathrm{363}}}+\sqrt{\frac{\mathrm{2}}{\mathrm{507}}}−\sqrt{\frac{\mathrm{2}}{\mathrm{675}}}+\sqrt{\frac{\mathrm{2}}{\mathrm{867}}}−._{} ... \\ $$

Question Number 117388 Answers: 0 Comments: 0

Question Number 117387 Answers: 1 Comments: 1

Solve the trigonometric equation 5sinθ+3=0 for value of θ from 0° to 360°

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{trigonometric}\:\mathrm{equation} \\ $$$$\mathrm{5sin}\theta+\mathrm{3}=\mathrm{0}\:\mathrm{for}\:\mathrm{value}\:\mathrm{of}\:\theta\:\mathrm{from}\:\mathrm{0}°\:\mathrm{to} \\ $$$$\mathrm{360}° \\ $$

Question Number 117380 Answers: 1 Comments: 1

... prove that ... Ω=∫_0 ^( ∞) (1/(2(√x)))sin(π^2 x+(1/x))dx=(1/( (√(8π)))) m.n.1970

$$\:\:\:\:\:\:\:\:\:\:\:...\:\:{prove}\:\:{that}\:... \\ $$$$\:\: \\ $$$$\Omega=\int_{\mathrm{0}} ^{\:\infty} \frac{\mathrm{1}}{\mathrm{2}\sqrt{{x}}}{sin}\left(\pi^{\mathrm{2}} {x}+\frac{\mathrm{1}}{{x}}\right){dx}=\frac{\mathrm{1}}{\:\sqrt{\mathrm{8}\pi}} \\ $$$$ \\ $$$$\:\:\:\:\:\:{m}.{n}.\mathrm{1970} \\ $$

Question Number 117371 Answers: 0 Comments: 0

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