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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 ^( (( π)/( 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 → R satisfy that (f ○ 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_(x→1) [x sin πx] (A) equals 1 (B) equals −1 (C) equals 0 (D) does not exist

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

Question Number 186752    Answers: 1   Comments: 0

lim_(x→+∞) (((√(x^3 −3x^2 +7))+((x^4 +3))^(1/3) )/( ((x^6 +2x^5 +1))^(1/4) −((x^7 +2x^3 +3))^(1/5) )) Please show work.

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

Question Number 186751    Answers: 3   Comments: 0

Question Number 186750    Answers: 1   Comments: 0

Question Number 186748    Answers: 0   Comments: 0

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

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

Question Number 186741    Answers: 1   Comments: 0

cos ((π/(18))).cos (((3π)/(18))).cos (((5π)/(18))).cos (((7π)/(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

Question Number 186726    Answers: 2   Comments: 0

Question Number 186721    Answers: 1   Comments: 1

Question Number 186705    Answers: 1   Comments: 1

a,b>0 , a+b=2 Prove that a^(2b) +b^(2a) +(((a−b)/2))^2 ≤2

$$\mathrm{a},\mathrm{b}>\mathrm{0}\:,\:\mathrm{a}+\mathrm{b}=\mathrm{2} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{a}^{\mathrm{2b}} +\mathrm{b}^{\mathrm{2a}} +\left(\frac{\mathrm{a}−\mathrm{b}}{\mathrm{2}}\right)^{\mathrm{2}} \leqslant\mathrm{2} \\ $$

Question Number 186701    Answers: 2   Comments: 2

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