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Question Number 118011 Answers: 4 Comments: 0

Question Number 118010 Answers: 3 Comments: 1

∫(dx/(x^4 +x^2 +1))

$$\int\frac{{dx}}{{x}^{\mathrm{4}} +{x}^{\mathrm{2}} +\mathrm{1}} \\ $$

Question Number 118005 Answers: 0 Comments: 1

Question Number 118004 Answers: 2 Comments: 0

find value of tan46^0 using calculus

$${find}\:\:{value}\:{of}\:{tan}\mathrm{46}^{\mathrm{0}} \:{using}\:{calculus} \\ $$

Question Number 118002 Answers: 1 Comments: 0

Question Number 117989 Answers: 1 Comments: 0

Question Number 117990 Answers: 1 Comments: 0

(2.3.5.7.9.11.13.17......∞)×(√(6/(3.8.24.48.80.120.168.288.....∞)))

$$\left(\mathrm{2}.\mathrm{3}.\mathrm{5}.\mathrm{7}.\mathrm{9}.\mathrm{11}.\mathrm{13}.\mathrm{17}......\infty\right)×\sqrt{\frac{\mathrm{6}}{\mathrm{3}.\mathrm{8}.\mathrm{24}.\mathrm{48}.\mathrm{80}.\mathrm{120}.\mathrm{168}.\mathrm{288}.....\infty}} \\ $$

Question Number 117984 Answers: 1 Comments: 0

If f(x) is a polynomial function satisfying the relation f(x)+f((1/x))=f(x)f((1/x)) for all 0≠x∈R and if f(2)=9, then f(6) is (A) 216 (B) 217 (C) 126 (D) 127

$$\mathrm{If}\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{polynomial}\:\mathrm{function}\:\mathrm{satisfying}\:\mathrm{the}\:\mathrm{relation} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{f}\left({x}\right)+{f}\left(\frac{\mathrm{1}}{{x}}\right)={f}\left({x}\right){f}\left(\frac{\mathrm{1}}{{x}}\right) \\ $$$$\mathrm{for}\:\mathrm{all}\:\mathrm{0}\neq{x}\in\mathbb{R}\:\mathrm{and}\:\mathrm{if}\:{f}\left(\mathrm{2}\right)=\mathrm{9},\:\mathrm{then}\:\mathrm{f}\left(\mathrm{6}\right)\:\mathrm{is} \\ $$$$\left(\mathrm{A}\right)\:\mathrm{216}\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{B}\right)\:\mathrm{217}\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{C}\right)\:\mathrm{126}\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:\mathrm{127} \\ $$

Question Number 117980 Answers: 0 Comments: 2

Where is the quiz?

$$\mathrm{Where}\:\mathrm{is}\:\mathrm{the}\:\mathrm{quiz}? \\ $$$$ \\ $$

Question Number 117979 Answers: 1 Comments: 1

... nice calculus... prove that: ∣ Γ ( i ) ∣=^? (√(π/(sinh(π)))) Γ: Euler gamma function ...m.n.july.1970...

$$\:\:\:\:\:\:\:\:\:\:...\:{nice}\:\:{calculus}... \\ $$$$\:\:\:\:\:{prove}\:\:{that}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\mid\:\:\Gamma\:\left(\:{i}\:\right)\:\mid\overset{?} {=}\:\sqrt{\frac{\pi}{{sinh}\left(\pi\right)}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\Gamma:\:\mathscr{E}{uler}\:{gamma}\:{function}\:\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{m}.{n}.{july}.\mathrm{1970}... \\ $$

Question Number 117973 Answers: 1 Comments: 0

consider a non−singular 2×2 square matrix T. If trace (T) =4 and trace (T^2 )=5 what is determinant of the matrix T ?

$$\mathrm{consider}\:\mathrm{a}\:\mathrm{non}−\mathrm{singular}\:\mathrm{2}×\mathrm{2}\: \\ $$$$\mathrm{square}\:\mathrm{matrix}\:\mathrm{T}.\:\mathrm{If}\:\mathrm{trace}\:\left(\mathrm{T}\right)\:=\mathrm{4} \\ $$$$\mathrm{and}\:\mathrm{trace}\:\left(\mathrm{T}^{\mathrm{2}} \right)=\mathrm{5}\:\mathrm{what}\:\mathrm{is}\:\mathrm{determinant} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{matrix}\:\mathrm{T}\:? \\ $$

Question Number 117972 Answers: 2 Comments: 0

The number of surjections of {1,2,3,4} onto {x,y} is (A) 16 (B) 8 (C) 14 (D) 6

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{surjections}\:\mathrm{of}\:\left\{\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4}\right\}\:\mathrm{onto}\:\left\{\mathrm{x},\mathrm{y}\right\}\:\mathrm{is} \\ $$$$\left(\mathrm{A}\right)\:\mathrm{16}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{B}\right)\:\mathrm{8}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{C}\right)\:\mathrm{14}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:\mathrm{6} \\ $$

Question Number 117963 Answers: 2 Comments: 0

Question Number 117960 Answers: 1 Comments: 0

Question Number 117948 Answers: 4 Comments: 0

... nice integral... please evaluate :: I =∫_0 ^( 1) (sin(x)+sin((1/x)))(dx/x) =?? m.n.1970

$$\:\:\:\:\:\:\:\:...\:\:{nice}\:\:{integral}...\: \\ $$$$\:\:\:{please}\:{evaluate}\::: \\ $$$$ \\ $$$$\:\:\:\:\mathrm{I}\:=\int_{\mathrm{0}} ^{\:\mathrm{1}} \left({sin}\left({x}\right)+{sin}\left(\frac{\mathrm{1}}{{x}}\right)\right)\frac{{dx}}{{x}}\:=?? \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:{m}.{n}.\mathrm{1970} \\ $$$$\:\: \\ $$

Question Number 117945 Answers: 3 Comments: 1

f(x) = ∫ ((5x^8 +7x^6 )/((2x^7 +x^2 +1)^2 )) dx and f(0) = 0 , then f(1) = _

$$\mathrm{f}\left(\mathrm{x}\right)\:=\:\int\:\frac{\mathrm{5x}^{\mathrm{8}} +\mathrm{7x}^{\mathrm{6}} }{\left(\mathrm{2x}^{\mathrm{7}} +\mathrm{x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} }\:\mathrm{dx}\:\mathrm{and}\:\: \\ $$$$\mathrm{f}\left(\mathrm{0}\right)\:=\:\mathrm{0}\:,\:\mathrm{then}\:\mathrm{f}\left(\mathrm{1}\right)\:=\:\_\: \\ $$

Question Number 117944 Answers: 3 Comments: 0

Find the value of k satisfies the equation ∫ _0^(π/3) (((tan x (√(cos x)))/( (√(2k)))) ) dx = 1−(1/( (√2)))

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{k}\:\mathrm{satisfies}\: \\ $$$$\mathrm{the}\:\mathrm{equation}\:\int\:_{\mathrm{0}} ^{\frac{\pi}{\mathrm{3}}} \:\left(\frac{\mathrm{tan}\:\mathrm{x}\:\sqrt{\mathrm{cos}\:\mathrm{x}}}{\:\sqrt{\mathrm{2k}}}\:\right)\:\mathrm{dx}\:=\:\mathrm{1}−\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}} \\ $$$$ \\ $$

Question Number 117943 Answers: 1 Comments: 0

solve (1/2)+((sin 112°)/(16sin 7°)) −cos 7°.cos 14°.cos 28°.cos 56° =?

$$\mathrm{solve}\:\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{sin}\:\mathrm{112}°}{\mathrm{16sin}\:\mathrm{7}°}\:−\mathrm{cos}\:\mathrm{7}°.\mathrm{cos}\:\mathrm{14}°.\mathrm{cos}\:\mathrm{28}°.\mathrm{cos}\:\mathrm{56}°\:=? \\ $$

Question Number 117953 Answers: 2 Comments: 0

... ◂nice integral▶... please prove : Ω=∫_0 ^( (π/2)) ln^2 (cot(x))dx =(π^3 /8) ...♠m.n.1070♠...

$$\:\:\:\:\:\:\:\:\:...\:\:\blacktriangleleft{nice}\:\:{integral}\blacktriangleright... \\ $$$$\:\:\:\:{please}\:\:{prove}\:: \\ $$$$ \\ $$$$\:\:\Omega=\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} {ln}^{\mathrm{2}} \left({cot}\left({x}\right)\right){dx}\:=\frac{\pi^{\mathrm{3}} }{\mathrm{8}}\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:...\spadesuit{m}.{n}.\mathrm{1070}\spadesuit... \\ $$

Question Number 117938 Answers: 5 Comments: 0

(1) ((∣a^2 −16∣)/(4−a)) − ((∣a^2 −9∣)/(3+a)) − ((∣4−a^2 ∣)/(2−a)) =? (2)(((∣x∣ +x )/(x−1)))^2 −((14x)/(x−1)) + 12 = 0 (3) log _4 (2^(2x) −(√3) cos x−6sin^2 x ) = x where ((5π)/2) ≤x≤4π (4) ((2cos^2 x−(√3) cos x)/(log _4 (sin x))) = 0 , where −3π ≤x≤−((3π)/2)

$$\left(\mathrm{1}\right)\:\frac{\mid{a}^{\mathrm{2}} −\mathrm{16}\mid}{\mathrm{4}−{a}}\:−\:\frac{\mid{a}^{\mathrm{2}} −\mathrm{9}\mid}{\mathrm{3}+{a}}\:−\:\frac{\mid\mathrm{4}−{a}^{\mathrm{2}} \mid}{\mathrm{2}−{a}}\:=? \\ $$$$\left(\mathrm{2}\right)\left(\frac{\mid\mathrm{x}\mid\:+\mathrm{x}\:}{\mathrm{x}−\mathrm{1}}\right)^{\mathrm{2}} −\frac{\mathrm{14x}}{\mathrm{x}−\mathrm{1}}\:+\:\mathrm{12}\:=\:\mathrm{0} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{log}\:_{\mathrm{4}} \left(\mathrm{2}^{\mathrm{2x}} −\sqrt{\mathrm{3}}\:\mathrm{cos}\:\mathrm{x}−\mathrm{6sin}\:^{\mathrm{2}} \mathrm{x}\:\right)\:=\:\mathrm{x} \\ $$$$\mathrm{where}\:\frac{\mathrm{5}\pi}{\mathrm{2}}\:\leqslant\mathrm{x}\leqslant\mathrm{4}\pi \\ $$$$\left(\mathrm{4}\right)\:\frac{\mathrm{2cos}\:^{\mathrm{2}} \mathrm{x}−\sqrt{\mathrm{3}}\:\mathrm{cos}\:\mathrm{x}}{\mathrm{log}\:_{\mathrm{4}} \left(\mathrm{sin}\:\mathrm{x}\right)}\:=\:\mathrm{0}\:,\:\mathrm{where}\: \\ $$$$−\mathrm{3}\pi\:\leqslant\mathrm{x}\leqslant−\frac{\mathrm{3}\pi}{\mathrm{2}} \\ $$

Question Number 117934 Answers: 1 Comments: 0

Let f : [1,∞)→[2,∞) be the function defined by f(x)=x+(1/x) If g : [2,∞)→[1,∞), is a function such that (g○f)(x)=x for all x≥1. Show that g(t)=((t+(√(t^2 −4)))/2)

$$\mathrm{Let}\:{f}\::\:\left[\mathrm{1},\infty\right)\rightarrow\left[\mathrm{2},\infty\right)\:\mathrm{be}\:\mathrm{the}\:\mathrm{function}\:\mathrm{defined}\:\mathrm{by} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{f}\left({x}\right)={x}+\frac{\mathrm{1}}{{x}} \\ $$$$\mathrm{If}\:\mathrm{g}\::\:\left[\mathrm{2},\infty\right)\rightarrow\left[\mathrm{1},\infty\right),\:\mathrm{is}\:\mathrm{a}\:\mathrm{function}\:\mathrm{such}\:\mathrm{that}\:\left(\mathrm{g}\circ{f}\right)\left({x}\right)={x} \\ $$$$\mathrm{for}\:\mathrm{all}\:{x}\geqslant\mathrm{1}.\:\mathrm{Show}\:\mathrm{that}\:\mathrm{g}\left({t}\right)=\frac{{t}+\sqrt{{t}^{\mathrm{2}} −\mathrm{4}}}{\mathrm{2}} \\ $$

Question Number 117926 Answers: 3 Comments: 0

tan^(−1) (((2x)/(x^2 −1))) + cot^(−1) (((x^2 −1)/(2x))) =((2π)/3) (tan^(−1) (x))^2 +(cot^(−1) (x))^2 =((5π)/8)

$$\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{2x}}{\mathrm{x}^{\mathrm{2}} −\mathrm{1}}\right)\:+\:\mathrm{cot}^{−\mathrm{1}} \left(\frac{\mathrm{x}^{\mathrm{2}} −\mathrm{1}}{\mathrm{2x}}\right)\:=\frac{\mathrm{2}\pi}{\mathrm{3}} \\ $$$$\left(\mathrm{tan}^{−\mathrm{1}} \left(\mathrm{x}\right)\right)^{\mathrm{2}} +\left(\mathrm{cot}^{−\mathrm{1}} \left(\mathrm{x}\right)\right)^{\mathrm{2}} =\frac{\mathrm{5}\pi}{\mathrm{8}} \\ $$$$ \\ $$

Question Number 117912 Answers: 1 Comments: 1

Question Number 117910 Answers: 3 Comments: 1

∫ sin^(−1) ((√x)) dx =?

$$\int\:\mathrm{sin}^{−\mathrm{1}} \left(\sqrt{\mathrm{x}}\right)\:\mathrm{dx}\:=? \\ $$

Question Number 117903 Answers: 2 Comments: 0

prove by mathematical induction that n(n+1)(n+2) is an integer multiple of 6

$${prove}\:{by}\:{mathematical}\:{induction} \\ $$$${that}\:{n}\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)\:{is}\:{an}\:{integer}\: \\ $$$${multiple}\:{of}\:\mathrm{6} \\ $$

Question Number 117901 Answers: 0 Comments: 1

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