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Question Number 97489    Answers: 2   Comments: 0

please prove it ∫_0 ^∞ e^(−ax^2 ) cos bx dx= (1/2)(√(π/a)).e^(−(b^2 /(4a)))

$${please}\:\:{prove}\:{it} \\ $$$$ \\ $$$$\int_{\mathrm{0}} ^{\infty} {e}^{−{ax}^{\mathrm{2}} } \mathrm{cos}\:{bx}\:\:{dx}=\:\frac{\mathrm{1}}{\mathrm{2}}\sqrt{\frac{\pi}{{a}}}.{e}^{−\frac{{b}^{\mathrm{2}} }{\mathrm{4}{a}}} \\ $$

Question Number 97485    Answers: 1   Comments: 3

Question Number 97483    Answers: 1   Comments: 4

In each week the growth of a plant is two−thirds the growth of the previous week. The plant grows 12 cm in the first week. (a) Calculate the growth of the plant in (b) the limiting height of the pant

$$\mathrm{In}\:\mathrm{each}\:\mathrm{week}\:\mathrm{the}\:\mathrm{growth}\:\mathrm{of}\:\mathrm{a}\:\mathrm{plant}\:\mathrm{is}\:\mathrm{two}−\mathrm{thirds} \\ $$$$\mathrm{the}\:\mathrm{growth}\:\mathrm{of}\:\mathrm{the}\:\mathrm{previous}\:\mathrm{week}. \\ $$$$\mathrm{The}\:\mathrm{plant}\:\mathrm{grows}\:\mathrm{12}\:\mathrm{cm}\:\mathrm{in}\:\mathrm{the}\:\mathrm{first}\:\mathrm{week}. \\ $$$$\left(\mathrm{a}\right)\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{growth}\:\mathrm{of}\:\mathrm{the}\:\mathrm{plant}\:\mathrm{in}\: \\ $$$$\left(\mathrm{b}\right)\:\mathrm{the}\:\mathrm{limiting}\:\mathrm{height}\:\mathrm{of}\:\mathrm{the}\:\mathrm{pant} \\ $$

Question Number 97479    Answers: 2   Comments: 2

Question Number 97478    Answers: 0   Comments: 0

Find the global parametrization of the curve { x^2 +y^2 +z^2 =1; x+y−z=0 }

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{global}\:\mathrm{parametrization} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{curve}\:\left\{\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{z}^{\mathrm{2}} =\mathrm{1};\:\mathrm{x}+\mathrm{y}−\mathrm{z}=\mathrm{0}\:\right\}\: \\ $$

Question Number 97476    Answers: 0   Comments: 0

2F1((1/2),(1/2);(1/2);z)=(1−z)^(1/2) ∗∗1 by kummer transformation 2F1((1/2),(1/2);(1/2);z)=2F1((1/2),(1/2);1+(1/2)+(1/2)−(1/2);z) 2F1((1/2),(1/2);(1/2);z)=((sin^(−1) (√(1−z)))/(√(1−z)))∗∗2 why do i get different answer in ∗∗1 and 2∗∗

$$\mathrm{2}{F}\mathrm{1}\left(\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{2}};\frac{\mathrm{1}}{\mathrm{2}};{z}\right)=\left(\mathrm{1}−{z}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \ast\ast\mathrm{1} \\ $$$${by}\:{kummer}\:{transformation} \\ $$$$\mathrm{2}{F}\mathrm{1}\left(\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{2}};\frac{\mathrm{1}}{\mathrm{2}};{z}\right)=\mathrm{2}{F}\mathrm{1}\left(\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{2}};\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{2}};{z}\right) \\ $$$$\mathrm{2}{F}\mathrm{1}\left(\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{2}};\frac{\mathrm{1}}{\mathrm{2}};{z}\right)=\frac{{sin}^{−\mathrm{1}} \sqrt{\mathrm{1}−{z}}}{\sqrt{\mathrm{1}−{z}}}\ast\ast\mathrm{2} \\ $$$$ \\ $$$${why}\:{do}\:{i}\:{get}\:{different}\:{answer}\:{in} \\ $$$$\ast\ast\mathrm{1}\:{and}\:\mathrm{2}\ast\ast \\ $$

Question Number 97465    Answers: 2   Comments: 1

∫_0 ^∝ e^(−x^4 ) dx=(1/4) please prove it

$$\int_{\mathrm{0}} ^{\propto} {e}^{−{x}^{\mathrm{4}} } {dx}=\frac{\mathrm{1}}{\mathrm{4}} \\ $$$${please}\:{prove}\:{it} \\ $$

Question Number 97463    Answers: 0   Comments: 2

Question Number 97462    Answers: 0   Comments: 1

Question Number 97460    Answers: 1   Comments: 2

Question Number 97454    Answers: 2   Comments: 0

Question Number 97439    Answers: 0   Comments: 2

∫_((√2)/2) ^1 ((x^3 /2) + (1/(6x)))(√(1+(((3x^2 )/2) −(1/(6x^2 )))^2 )) dx

$$\underset{\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}} {\overset{\mathrm{1}} {\int}}\:\left(\frac{\mathrm{x}^{\mathrm{3}} }{\mathrm{2}}\:+\:\frac{\mathrm{1}}{\mathrm{6x}}\right)\sqrt{\mathrm{1}+\left(\frac{\mathrm{3x}^{\mathrm{2}} }{\mathrm{2}}\:−\frac{\mathrm{1}}{\mathrm{6x}^{\mathrm{2}} }\right)^{\mathrm{2}} }\:\:\mathrm{dx} \\ $$

Question Number 97438    Answers: 1   Comments: 1

If −3≤x≤4, −2≤y≤5, 4≤z≤10 , find the greatest value of w = z−xy

$$\mathrm{If}\:−\mathrm{3}\leqslant\mathrm{x}\leqslant\mathrm{4},\:−\mathrm{2}\leqslant\mathrm{y}\leqslant\mathrm{5},\:\mathrm{4}\leqslant\mathrm{z}\leqslant\mathrm{10} \\ $$$$,\:\mathrm{find}\:\mathrm{the}\:\mathrm{greatest} \\ $$$$\mathrm{value}\:\mathrm{of}\:\mathrm{w}\:=\:\mathrm{z}−\mathrm{xy}\: \\ $$

Question Number 97423    Answers: 2   Comments: 0

solve y^(′′) +4y =xe^(−x) with y(0)=1 and y^′ (0) =−1

$$\mathrm{solve}\:\mathrm{y}^{''} \:+\mathrm{4y}\:=\mathrm{xe}^{−\mathrm{x}} \:\:\:\:\mathrm{with}\:\:\mathrm{y}\left(\mathrm{0}\right)=\mathrm{1}\:\mathrm{and}\:\mathrm{y}^{'} \left(\mathrm{0}\right)\:=−\mathrm{1} \\ $$

Question Number 97428    Answers: 2   Comments: 5

Question Number 97418    Answers: 1   Comments: 1

Verify if the series Σ_(n=1) ^n ((2n + 5)/(n^2 +3n + 2)) is convergent or divergent. What method is easier?

$$\mathrm{Verify}\:\mathrm{if}\:\mathrm{the}\:\mathrm{series}\: \\ $$$$\:\underset{{n}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{2}{n}\:+\:\mathrm{5}}{{n}^{\mathrm{2}} \:+\mathrm{3}{n}\:+\:\mathrm{2}}\:\mathrm{is}\:\mathrm{convergent}\:\mathrm{or}\:\mathrm{divergent}. \\ $$$$\mathrm{What}\:\mathrm{method}\:\mathrm{is}\:\mathrm{easier}? \\ $$

Question Number 97417    Answers: 1   Comments: 0

∫(x/(a+sin^2 x))dx=?

$$\int\frac{{x}}{{a}+\mathrm{sin}^{\mathrm{2}} \:{x}}{dx}=? \\ $$

Question Number 97413    Answers: 1   Comments: 0

Given that ω = e^(iθ) , θ≠ nπ , n ∈N show that (1 + ω)^n = 2^n ((1/2)θ)e^((1/2)(inθ)) please help me out on this, i′ve stumbled on it.

$$\mathrm{Given}\:\mathrm{that}\:\omega\:=\:{e}^{{i}\theta} ,\:\theta\neq\:{n}\pi\:,\:{n}\:\in\mathbb{N} \\ $$$$\mathrm{show}\:\mathrm{that}\:\left(\mathrm{1}\:+\:\omega\right)^{{n}} \:=\:\mathrm{2}^{{n}} \left(\frac{\mathrm{1}}{\mathrm{2}}\theta\right){e}^{\frac{\mathrm{1}}{\mathrm{2}}\left({in}\theta\right)} \\ $$$$\mathrm{please}\:\mathrm{help}\:\mathrm{me}\:\mathrm{out}\:\mathrm{on}\:\mathrm{this},\:\mathrm{i}'\mathrm{ve}\:\mathrm{stumbled}\:\mathrm{on}\:\mathrm{it}. \\ $$

Question Number 97412    Answers: 2   Comments: 0

prove (√2)<log_2 3<(√3)

$${prove} \\ $$$$\sqrt{\mathrm{2}}<\mathrm{log}_{\mathrm{2}} \:\mathrm{3}<\sqrt{\mathrm{3}} \\ $$

Question Number 97403    Answers: 0   Comments: 1

Question Number 97401    Answers: 0   Comments: 1

Question Number 97400    Answers: 0   Comments: 2

∫((xdx)/(sin^2 x−3))=? help me

$$\int\frac{\mathrm{xdx}}{\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}−\mathrm{3}}=? \\ $$$$\:\:\mathrm{help}\:\mathrm{me} \\ $$

Question Number 97398    Answers: 0   Comments: 0

The discontinuty of [x]^2 −[x^2 ]

$$\mathrm{The}\:\mathrm{discontinuty}\:\mathrm{of}\:\left[\mathrm{x}\right]^{\mathrm{2}} −\left[\mathrm{x}^{\mathrm{2}} \right] \\ $$

Question Number 97390    Answers: 1   Comments: 0

compare log_2 3 with log_3 4

$${compare}\:{log}_{\mathrm{2}} \mathrm{3}\:{with}\:{log}_{\mathrm{3}} \mathrm{4} \\ $$

Question Number 97386    Answers: 0   Comments: 2

Question Number 97380    Answers: 0   Comments: 0

E is a vectorial plane. his base is B=(i^→ ;j^→ ). f is an endomorphism defined by f(i^→ )=−((√2)/2)i^→ +((√2)/2)j^→ and f(j^→ )=((√2)/2)i^→ −((√2)/2)j^→ 1)Show that ker f is a vectorial straigh line and his base is e_1 ^→ =(√2)i^→ +(√2)j^→ 2)show that G, the set of vectors u^→ ∈ E such as f(u^→ )=(√2)u^→ is a vectorial straigh line and his Base is e_(2 ) ^→ =i^→ +j^→ 3) Determine the matrix A′ of f in B′ if B′=(e_1 ^→ ;e_2 ^→ ).

$${E}\:{is}\:{a}\:{vectorial}\:{plane}.\:{his}\:{base}\:{is}\: \\ $$$${B}=\left(\overset{\rightarrow} {{i}};\overset{\rightarrow} {{j}}\right).\:{f}\:{is}\:{an}\:{endomorphism}\:{defined} \\ $$$${by}\:{f}\left(\overset{\rightarrow} {{i}}\right)=−\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\overset{\rightarrow} {{i}}+\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\overset{\rightarrow} {{j}}\:{and}\:{f}\left(\overset{\rightarrow} {{j}}\right)=\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\overset{\rightarrow} {{i}}−\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\overset{\rightarrow} {{j}} \\ $$$$\left.\mathrm{1}\right){Show}\:{that}\:{ker}\:{f}\:{is}\:{a}\:{vectorial}\:{straigh} \\ $$$${line}\:{and}\:{his}\:{base}\:{is}\:\overset{\rightarrow} {{e}}_{\mathrm{1}} =\sqrt{\mathrm{2}}\overset{\rightarrow} {{i}}+\sqrt{\mathrm{2}}\overset{\rightarrow} {{j}} \\ $$$$\left.\mathrm{2}\right){show}\:{that}\:{G},\:{the}\:{set}\:{of}\:{vectors}\:\overset{\rightarrow} {{u}} \\ $$$$\:\in\:{E}\:{such}\:{as}\:{f}\left(\overset{\rightarrow} {{u}}\right)=\sqrt{\mathrm{2}}\overset{\rightarrow} {{u}}\:{is}\:{a}\:{vectorial}\:{straigh} \\ $$$${line}\:{and}\:{his}\:{Base}\:{is}\:\overset{\rightarrow} {{e}}_{\mathrm{2}\:\:} =\overset{\rightarrow} {{i}}+\overset{\rightarrow} {{j}} \\ $$$$\left.\mathrm{3}\right)\:{Determine}\:{the}\:{matrix}\:{A}'\:{of}\:{f}\:{in} \\ $$$${B}'\:{if}\:{B}'=\left(\overset{\rightarrow} {{e}}_{\mathrm{1}} ;\overset{\rightarrow} {{e}}_{\mathrm{2}} \right). \\ $$

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