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Question Number 186803    Answers: 1   Comments: 0

Question Number 186800    Answers: 0   Comments: 0

show that ∫_0 ^( ∞ ) ((tan^(βˆ’1) 𝛂x tan^(βˆ’1) 𝛃x)/x^2 )dx = (𝛑/2)log{(((𝛂+𝛃)^(𝛂+𝛃) )/(𝛂^𝛂 𝛃^𝛃 ))}

$$\boldsymbol{\mathrm{show}}\:\boldsymbol{\mathrm{that}} \\ $$$$\int_{\mathrm{0}} ^{\:\infty\:} \frac{\boldsymbol{\mathrm{tan}}^{βˆ’\mathrm{1}} \boldsymbol{\alpha{x}}\:\boldsymbol{\mathrm{tan}}^{βˆ’\mathrm{1}} \boldsymbol{\beta{x}}}{\boldsymbol{{x}}^{\mathrm{2}} }\boldsymbol{{dx}}\:=\:\frac{\boldsymbol{\pi}}{\mathrm{2}}\mathrm{log}\left\{\frac{\left(\boldsymbol{\alpha}+\boldsymbol{\beta}\right)^{\boldsymbol{\alpha}+\boldsymbol{\beta}} }{\boldsymbol{\alpha}^{\boldsymbol{\alpha}} \boldsymbol{\beta}^{\boldsymbol{\beta}} }\right\}\: \\ $$

Question Number 186790    Answers: 0   Comments: 0

Solve using Fourierβ€²s series: x^2 (1βˆ’x)yβ€²β€²βˆ’x(1+x)yβ€²+y=0

$${Solve}\:{using}\:{Fourier}'{s}\:{series}: \\ $$$${x}^{\mathrm{2}} \left(\mathrm{1}βˆ’{x}\right){y}''βˆ’{x}\left(\mathrm{1}+{x}\right){y}'+{y}=\mathrm{0} \\ $$

Question Number 186788    Answers: 1   Comments: 0

Question Number 186787    Answers: 0   Comments: 0

Question Number 186786    Answers: 0   Comments: 0

Question Number 186785    Answers: 0   Comments: 0

Question Number 186784    Answers: 1   Comments: 0

Question Number 186783    Answers: 0   Comments: 0

Question Number 186782    Answers: 3   Comments: 0

Question Number 186772    Answers: 0   Comments: 0

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Question Number 186771    Answers: 2   Comments: 0

Q : Find the value of the following integral. I = ∫_0 ^( (( Ο€)/( 2))) (( 1)/( 1 + sin^( 4) ( x ) + cos^( 4) ( x ) )) dx = ?

$$ \\ $$$$\:\:\:\:\mathrm{Q}\::\:\:\:\:\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{integral}.\:\:\:\:\:\:\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{I}\:=\:\int_{\mathrm{0}} ^{\:\frac{\:\pi}{\:\mathrm{2}}} \:\frac{\:\:\mathrm{1}}{\:\mathrm{1}\:+\:\mathrm{sin}^{\:\mathrm{4}} \:\left(\:{x}\:\right)\:+\:\mathrm{cos}^{\:\mathrm{4}} \:\left(\:{x}\:\right)\:}\:\mathrm{d}{x}\:=\:\:?\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 186780    Answers: 1   Comments: 0

∫ ((1 + sin x + cos x)/(1 + sin x)) dx

$$ \\ $$$$\:\:\:\:\:\:\:\int\:\:\:\frac{\mathrm{1}\:+\:\boldsymbol{{sin}}\:\boldsymbol{{x}}\:+\:\boldsymbol{{cos}}\:\boldsymbol{{x}}}{\mathrm{1}\:+\:\boldsymbol{{sin}}\:\boldsymbol{{x}}}\:\:\boldsymbol{{dx}}\:\: \\ $$$$ \\ $$

Question Number 186769    Answers: 0   Comments: 0

Given function f : R β†’ R satisfy that (f β—‹ f)(x) + x = (x+1) f(x) . Find the value of f(1) .

$$\mathrm{Given}\:\:\mathrm{function}\:\:\:{f}\::\:\mathbb{R}\:\rightarrow\:\mathbb{R}\:\:\:\mathrm{satisfy}\:\:\mathrm{that} \\ $$$$\:\:\:\:\left({f}\:\circ\:{f}\right)\left({x}\right)\:+\:{x}\:=\:\left({x}+\mathrm{1}\right)\:{f}\left({x}\right)\:. \\ $$$$\mathrm{Find}\:\:\mathrm{the}\:\:\mathrm{value}\:\:\mathrm{of}\:\:{f}\left(\mathrm{1}\right)\:. \\ $$

Question Number 186764    Answers: 0   Comments: 3

Question Number 186762    Answers: 3   Comments: 4

Question Number 186758    Answers: 2   Comments: 0

If [t] denotes the integral part of t, then lim_(xβ†’1) [x sin Ο€x] (A) equals 1 (B) equals βˆ’1 (C) equals 0 (D) does not exist

$$\mathrm{If}\:\left[{t}\right]\:\mathrm{denotes}\:\mathrm{the}\:\mathrm{integral}\:\mathrm{part}\:\mathrm{of}\:{t},\:\mathrm{then}\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\left[{x}\:\mathrm{sin}\:\pi{x}\right] \\ $$$$\left(\mathrm{A}\right)\:\:\mathrm{equals}\:\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{B}\right)\:\:\mathrm{equals}\:βˆ’\mathrm{1} \\ $$$$\left(\mathrm{C}\right)\:\:\mathrm{equals}\:\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:\mathrm{does}\:\mathrm{not}\:\mathrm{exist} \\ $$

Question Number 186752    Answers: 1   Comments: 0

lim_(xβ†’+∞) (((√(x^3 βˆ’3x^2 +7))+((x^4 +3))^(1/3) )/( ((x^6 +2x^5 +1))^(1/4) βˆ’((x^7 +2x^3 +3))^(1/5) )) Please show work.

$$\underset{{x}\rightarrow+\infty} {\mathrm{lim}}\:\frac{\sqrt{{x}^{\mathrm{3}} βˆ’\mathrm{3}{x}^{\mathrm{2}} +\mathrm{7}}+\sqrt[{\mathrm{3}}]{{x}^{\mathrm{4}} +\mathrm{3}}}{\:\sqrt[{\mathrm{4}}]{{x}^{\mathrm{6}} +\mathrm{2}{x}^{\mathrm{5}} +\mathrm{1}}βˆ’\sqrt[{\mathrm{5}}]{{x}^{\mathrm{7}} +\mathrm{2}{x}^{\mathrm{3}} +\mathrm{3}}} \\ $$$${Please}\:{show}\:{work}. \\ $$

Question Number 186751    Answers: 3   Comments: 0

Question Number 186750    Answers: 1   Comments: 0

Question Number 186748    Answers: 0   Comments: 0

Let f:R^+ β†’R^+ be a function satisfying the relation f(x.f(y))=f(xy)+x for all x, y ∈R^+ . Then lim_(xβ†’0) ((((f(x))^(1/3) βˆ’1)/((f(x))^(1/2) βˆ’1)))= (A) 1 (B) (1/2) (C) (2/3) (D) (3/2)

$$\mathrm{Let}\:{f}:\mathbb{R}^{+} \rightarrow\mathbb{R}^{+} \:\mathrm{be}\:\mathrm{a}\:\mathrm{function}\:\mathrm{satisfying}\:\mathrm{the}\:\mathrm{relation} \\ $$$${f}\left({x}.{f}\left(\mathrm{y}\right)\right)={f}\left({x}\mathrm{y}\right)+{x}\:\mathrm{for}\:\mathrm{all}\:{x},\:\mathrm{y}\:\in\mathbb{R}^{+} .\:\mathrm{Then} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\left({f}\left({x}\right)\right)^{\mathrm{1}/\mathrm{3}} βˆ’\mathrm{1}}{\left({f}\left({x}\right)\right)^{\mathrm{1}/\mathrm{2}} βˆ’\mathrm{1}}\right)= \\ $$$$\left(\mathrm{A}\right)\:\:\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{B}\right)\:\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\left(\mathrm{C}\right)\:\:\frac{\mathrm{2}}{\mathrm{3}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:\:\frac{\mathrm{3}}{\mathrm{2}} \\ $$

Question Number 186741    Answers: 1   Comments: 0

cos ((Ο€/(18))).cos (((3Ο€)/(18))).cos (((5Ο€)/(18))).cos (((7Ο€)/(18)))=?

$$\:\mathrm{cos}\:\left(\frac{\pi}{\mathrm{18}}\right).\mathrm{cos}\:\left(\frac{\mathrm{3}\pi}{\mathrm{18}}\right).\mathrm{cos}\:\left(\frac{\mathrm{5}\pi}{\mathrm{18}}\right).\mathrm{cos}\:\left(\frac{\mathrm{7}\pi}{\mathrm{18}}\right)=? \\ $$

Question Number 186739    Answers: 1   Comments: 0

((5+((5+((5+...))^(1/3) ))^(1/3) ))^(1/3) =?

$$\sqrt[{\mathrm{3}}]{\mathrm{5}+\sqrt[{\mathrm{3}}]{\mathrm{5}+\sqrt[{\mathrm{3}}]{\mathrm{5}+...}}}=? \\ $$

Question Number 186737    Answers: 1   Comments: 1

Question Number 186736    Answers: 1   Comments: 0

Question Number 186735    Answers: 1   Comments: 0

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