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Question Number 98589 Answers: 1 Comments: 0

calculate ∫_0 ^∞ ((sin(αx^2 ))/(x^2 +4))dx with α real

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{sin}\left(\alpha\mathrm{x}^{\mathrm{2}} \right)}{\mathrm{x}^{\mathrm{2}} \:+\mathrm{4}}\mathrm{dx}\:\:\mathrm{with}\:\alpha\:\mathrm{real} \\ $$

Question Number 98588 Answers: 0 Comments: 0

calculate ∫_(−∞) ^∞ ((xsin(x))/((x^2 +x+1)^2 ))dx

$$\mathrm{calculate}\:\int_{−\infty} ^{\infty} \:\frac{\mathrm{xsin}\left(\mathrm{x}\right)}{\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} }\mathrm{dx} \\ $$

Question Number 98587 Answers: 1 Comments: 0

calculate ∫_(−∞) ^(+∞) ((cos(αx))/(x^4 +1))dx (α real)

$$\mathrm{calculate}\:\:\int_{−\infty} ^{+\infty} \:\frac{\mathrm{cos}\left(\alpha\mathrm{x}\right)}{\mathrm{x}^{\mathrm{4}} \:+\mathrm{1}}\mathrm{dx}\:\:\left(\alpha\:\mathrm{real}\right) \\ $$

Question Number 98575 Answers: 0 Comments: 6

Calculate the force F needed to punch at 1.46cm diametre hole in a steel plate 1.27cm tjick. tbe ultimate shear stress of the steel is 3.45×10^(8N/m^2 )

$$\mathrm{Calculate}\:\mathrm{the}\:\mathrm{force}\:\mathrm{F}\:\mathrm{needed}\:\mathrm{to}\:\mathrm{punch}\:\mathrm{at}\:\mathrm{1}.\mathrm{46cm}\:\mathrm{diametre}\:\mathrm{hole}\:\mathrm{in}\:\mathrm{a}\:\mathrm{steel}\:\mathrm{plate}\:\mathrm{1}.\mathrm{27cm}\:\mathrm{tjick}.\:\mathrm{tbe}\:\mathrm{ultimate}\:\mathrm{shear}\:\mathrm{stress}\:\mathrm{of}\:\mathrm{the}\:\mathrm{steel}\:\mathrm{is}\:\mathrm{3}.\mathrm{45}×\mathrm{10}^{\mathrm{8N}/\mathrm{m}^{\mathrm{2}} } \\ $$

Question Number 98570 Answers: 1 Comments: 1

a,b,c>0 prove: (a/(√(a^2 +8bc)))+(b/(√(b^2 +8ac)))+(c/(√(c^2 +8ab)))≥1 help please...

$$\boldsymbol{{a}},\boldsymbol{{b}},\boldsymbol{{c}}>\mathrm{0}\:\:\:\:\:\:\:\boldsymbol{{prove}}: \\ $$$$\frac{\boldsymbol{{a}}}{\sqrt{\boldsymbol{{a}}^{\mathrm{2}} +\mathrm{8}\boldsymbol{{bc}}}}+\frac{\boldsymbol{{b}}}{\sqrt{\boldsymbol{{b}}^{\mathrm{2}} +\mathrm{8}\boldsymbol{{ac}}}}+\frac{\boldsymbol{{c}}}{\sqrt{\boldsymbol{{c}}^{\mathrm{2}} +\mathrm{8}\boldsymbol{{ab}}}}\geqslant\mathrm{1} \\ $$$$\boldsymbol{{help}}\:\boldsymbol{{please}}... \\ $$

Question Number 98568 Answers: 1 Comments: 1

Question Number 98560 Answers: 2 Comments: 1

Question Number 98557 Answers: 0 Comments: 0

a_n =Σ_(k=1 ) ^(n−1) ((sin((((2k−1)π)/(2n))))/(cos^2 ((((k−1)π)/(2n)))cos^2 (((kπ)/(2n))))) find lim_(n→∞) (a_n /n^3 )

$${a}_{{n}} =\underset{{k}=\mathrm{1}\:} {\overset{{n}−\mathrm{1}} {\sum}}\frac{{sin}\left(\frac{\left(\mathrm{2}{k}−\mathrm{1}\right)\pi}{\mathrm{2}{n}}\right)}{{cos}^{\mathrm{2}} \left(\frac{\left({k}−\mathrm{1}\right)\pi}{\mathrm{2}{n}}\right){cos}^{\mathrm{2}} \left(\frac{{k}\pi}{\mathrm{2}{n}}\right)} \\ $$$$ \\ $$$${find}\:\:\underset{{n}\rightarrow\infty} {{lim}}\frac{{a}_{{n}} }{{n}^{\mathrm{3}} } \\ $$

Question Number 98544 Answers: 1 Comments: 1

Question Number 98539 Answers: 2 Comments: 0

Given the function f(x) = ((ln x)/(x−1)) (a) State the domain D_f of f. (b) Find lim_(x→∞) ((ln x)/(x−1)). State its asymptotes. (c) Draw up a variation table for the curve y = f(x).

$$\mathrm{Given}\:\mathrm{the}\:\mathrm{function} \\ $$$${f}\left({x}\right)\:=\:\frac{\mathrm{ln}\:{x}}{{x}−\mathrm{1}} \\ $$$$\left(\mathrm{a}\right)\:\mathrm{State}\:\mathrm{the}\:\mathrm{domain}\:{D}_{{f}} \:\mathrm{of}\:{f}. \\ $$$$\left(\mathrm{b}\right)\:\mathrm{Find}\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{ln}\:{x}}{{x}−\mathrm{1}}.\:\mathrm{State}\:\mathrm{its}\:\mathrm{asymptotes}. \\ $$$$\left(\mathrm{c}\right)\:\mathrm{Draw}\:\mathrm{up}\:\mathrm{a}\:\mathrm{variation}\:\mathrm{table}\:\mathrm{for}\:\mathrm{the}\:\mathrm{curve}\:{y}\:=\:{f}\left({x}\right).\: \\ $$

Question Number 98537 Answers: 1 Comments: 1

Question Number 98535 Answers: 0 Comments: 0

f(x) = log _5 (x) + 5e^(3x) f^(−1) (x) = ?

$${f}\left({x}\right)\:=\:\mathrm{log}\:_{\mathrm{5}} \left({x}\right)\:+\:\mathrm{5}{e}^{\mathrm{3}{x}} \\ $$$${f}^{−\mathrm{1}} \left({x}\right)\:=\:? \\ $$

Question Number 98531 Answers: 0 Comments: 0

Question Number 98528 Answers: 0 Comments: 2

Question Number 98521 Answers: 1 Comments: 0

Question Number 98520 Answers: 1 Comments: 0

Integrate the function f(x,y) = xy(x^2 +y^2 ) over the domain R:{−3≤x^2 −y^2 ≤3, 1≤xy≤4}

$$\mathrm{Integrate}\:\mathrm{the}\:\mathrm{function}\:\mathrm{f}\left(\mathrm{x},\mathrm{y}\right)\:=\:\mathrm{xy}\left(\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} \right) \\ $$$$\mathrm{over}\:\mathrm{the}\:\mathrm{domain}\:\mathrm{R}:\left\{−\mathrm{3}\leqslant\mathrm{x}^{\mathrm{2}} −\mathrm{y}^{\mathrm{2}} \leqslant\mathrm{3},\:\mathrm{1}\leqslant\mathrm{xy}\leqslant\mathrm{4}\right\} \\ $$

Question Number 98516 Answers: 2 Comments: 2

Question Number 98506 Answers: 0 Comments: 1

Question Number 98470 Answers: 0 Comments: 0

prove that asymtotes y=mx−((∂xφ_K )/φ_n ) k=n−1 cuts the curve Σ_(r=0) ^n φ_r ((y/x))x^r in n(n−1) points

$${prove}\:{that}\:{asymtotes}\: \\ $$$${y}={mx}−\frac{\partial{x}\phi_{{K}} }{\phi_{{n}} } \\ $$$${k}={n}−\mathrm{1} \\ $$$$ \\ $$$${cuts}\:{the}\:{curve}\: \\ $$$$\underset{{r}=\mathrm{0}} {\overset{{n}} {\sum}}\phi_{{r}} \left(\frac{{y}}{{x}}\right){x}^{{r}} \\ $$$${in}\:{n}\left({n}−\mathrm{1}\right)\:{points} \\ $$$$ \\ $$

Question Number 98468 Answers: 2 Comments: 10

Question Number 98464 Answers: 1 Comments: 0

Working individually, A, B and C can finish a piece of work in 16 days, 20 days and 30 days respectively. In how many days can A, B and C together complete a work which is 3(1/2) times the previous work?

$$\mathrm{Working}\:\mathrm{individually},\:\mathrm{A},\:\mathrm{B}\:\mathrm{and}\:\mathrm{C}\:\mathrm{can} \\ $$$$\mathrm{finish}\:\mathrm{a}\:\mathrm{piece}\:\mathrm{of}\:\mathrm{work}\:\mathrm{in}\:\mathrm{16}\:\mathrm{days},\:\mathrm{20}\:\mathrm{days} \\ $$$$\mathrm{and}\:\mathrm{30}\:\mathrm{days}\:\mathrm{respectively}.\:\mathrm{In}\:\mathrm{how}\:\mathrm{many} \\ $$$$\mathrm{days}\:\mathrm{can}\:\mathrm{A},\:\mathrm{B}\:\mathrm{and}\:\mathrm{C}\:\mathrm{together}\:\mathrm{complete} \\ $$$$\mathrm{a}\:\mathrm{work}\:\mathrm{which}\:\mathrm{is}\:\mathrm{3}\frac{\mathrm{1}}{\mathrm{2}}\:\mathrm{times}\:\mathrm{the}\:\mathrm{previous} \\ $$$$\mathrm{work}? \\ $$

Question Number 98463 Answers: 1 Comments: 0

Question Number 98461 Answers: 3 Comments: 1

Question Number 98453 Answers: 0 Comments: 2

(m/(1×2))×(m/(2×3))×(m/(3×4))×...×(m/(99×100))=99 m?

$$\frac{{m}}{\mathrm{1}×\mathrm{2}}×\frac{{m}}{\mathrm{2}×\mathrm{3}}×\frac{{m}}{\mathrm{3}×\mathrm{4}}×...×\frac{{m}}{\mathrm{99}×\mathrm{100}}=\mathrm{99} \\ $$$$ \\ $$$${m}? \\ $$

Question Number 98613 Answers: 0 Comments: 0

Question Number 98450 Answers: 0 Comments: 1

Find the supremum and the infimum of f(x) = (x/(sin x)) ,x∈ (0,(π/2) ]

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{supremum}\:\mathrm{and}\:\mathrm{the} \\ $$$$\mathrm{infimum}\:\mathrm{of}\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\frac{\mathrm{x}}{\mathrm{sin}\:\mathrm{x}}\:,\mathrm{x}\in\:\left(\mathrm{0},\frac{\pi}{\mathrm{2}}\:\right] \\ $$

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