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How do you express this question in partial fraction ((5x^2 +x+6)/((3−2x)(x^2 +4))) hence obtain the expansion is ascending powers of x up to and including the term x^2

$${How}\:{do}\:{you}\:{express}\:{this}\:{question} \\ $$$${in}\:{partial}\:{fraction}\:\frac{\mathrm{5}{x}^{\mathrm{2}} +{x}+\mathrm{6}}{\left(\mathrm{3}−\mathrm{2}{x}\right)\left({x}^{\mathrm{2}} +\mathrm{4}\right)}\: \\ $$$${hence}\:{obtain}\:{the}\:{expansion} \\ $$$${is}\:{ascending}\:{powers}\:{of}\:{x}\:{up} \\ $$$${to}\:{and}\:{including}\:{the}\:{term}\:{x}^{\mathrm{2}} \\ $$

Question Number 118386 Answers: 1 Comments: 0

if:x+(√y)=a , (√x)+y=b find x,y

$${if}:{x}+\sqrt{{y}}={a}\:,\:\sqrt{{x}}+{y}={b}\:{find}\:{x},{y} \\ $$

Question Number 118384 Answers: 1 Comments: 0

Question Number 118376 Answers: 2 Comments: 0

If a curve y = f(x) passing through the point (1,2) is the solution of differential equation 2x^2 dy = (2xy+y^2 )dx , then the value of f(2) is equal to?

$${If}\:{a}\:{curve}\:{y}\:=\:{f}\left({x}\right)\:{passing}\:{through} \\ $$$${the}\:{point}\:\left(\mathrm{1},\mathrm{2}\right)\:{is}\:{the}\:{solution} \\ $$$${of}\:{differential}\:{equation} \\ $$$$\mathrm{2}{x}^{\mathrm{2}} \:{dy}\:=\:\left(\mathrm{2}{xy}+{y}^{\mathrm{2}} \right){dx}\:,\:{then}\:{the}\: \\ $$$${value}\:{of}\:{f}\left(\mathrm{2}\right)\:{is}\:{equal}\:{to}? \\ $$

Question Number 207620 Answers: 1 Comments: 0

prove that ∫_(−a) ^a (dx/(x^n +1+(√(x^(2n) +1))))=a

$$\mathrm{prove}\:\mathrm{that}\:\underset{−{a}} {\overset{{a}} {\int}}\:\frac{{dx}}{{x}^{{n}} +\mathrm{1}+\sqrt{{x}^{\mathrm{2}{n}} +\mathrm{1}}}={a} \\ $$

Question Number 118371 Answers: 0 Comments: 1

old problem question 118120 tan tan x =tan 3x −tan 2x let t=tan x (1) tan t =((t^5 +2t^3 +t)/(3t^4 −4t^2 +1)) for t≥0 we get (approximating) t_0 =0 t_1 ≈1.28941477 t_2 ≈4.17629616 t_3 ≈7.49316173 t_4 ≈10.7303610 t_5 ≈13.9285293 ... x=nπ+arctan t let n=0 to stay in the first period 0≤t<+∞ ⇒ 0≤arctan t <(π/2) ⇒ (1) has infinite solutions for 0≤x<(π/2) graphically this is easy to see, plot these: f_1 (t)=tan t f_2 (t)=((t(t^4 +2t^2 +1))/(3t^4 −4t^2 +1))=(t/3)+((2t(5t^2 +1))/(3(3t^4 −4t^2 +1))) ⇒ g(t)=(1/3)t is asymptote of f_1 (t) and obviously tan t =at with a∈R has infinite solutions

$$\mathrm{old}\:\mathrm{problem}\:{question}\:\mathrm{118120} \\ $$$$\mathrm{tan}\:\mathrm{tan}\:{x}\:=\mathrm{tan}\:\mathrm{3}{x}\:−\mathrm{tan}\:\mathrm{2}{x} \\ $$$$\mathrm{let}\:{t}=\mathrm{tan}\:{x} \\ $$$$\left(\mathrm{1}\right)\:\:\:\:\:\mathrm{tan}\:{t}\:=\frac{{t}^{\mathrm{5}} +\mathrm{2}{t}^{\mathrm{3}} +{t}}{\mathrm{3}{t}^{\mathrm{4}} −\mathrm{4}{t}^{\mathrm{2}} +\mathrm{1}} \\ $$$$\mathrm{for}\:{t}\geqslant\mathrm{0}\:\mathrm{we}\:\mathrm{get}\:\left(\mathrm{approximating}\right) \\ $$$${t}_{\mathrm{0}} =\mathrm{0} \\ $$$${t}_{\mathrm{1}} \approx\mathrm{1}.\mathrm{28941477} \\ $$$${t}_{\mathrm{2}} \approx\mathrm{4}.\mathrm{17629616} \\ $$$${t}_{\mathrm{3}} \approx\mathrm{7}.\mathrm{49316173} \\ $$$${t}_{\mathrm{4}} \approx\mathrm{10}.\mathrm{7303610} \\ $$$${t}_{\mathrm{5}} \approx\mathrm{13}.\mathrm{9285293} \\ $$$$... \\ $$$${x}={n}\pi+\mathrm{arctan}\:{t} \\ $$$$\mathrm{let}\:{n}=\mathrm{0}\:\mathrm{to}\:\mathrm{stay}\:\mathrm{in}\:\mathrm{the}\:\mathrm{first}\:\mathrm{period} \\ $$$$\mathrm{0}\leqslant{t}<+\infty\:\Rightarrow\:\mathrm{0}\leqslant\mathrm{arctan}\:{t}\:<\frac{\pi}{\mathrm{2}} \\ $$$$\Rightarrow\:\left(\mathrm{1}\right)\:\mathrm{has}\:\mathrm{infinite}\:\mathrm{solutions}\:\mathrm{for}\:\mathrm{0}\leqslant{x}<\frac{\pi}{\mathrm{2}} \\ $$$$\mathrm{graphically}\:\mathrm{this}\:\mathrm{is}\:\mathrm{easy}\:\mathrm{to}\:\mathrm{see},\:\mathrm{plot}\:\mathrm{these}: \\ $$$${f}_{\mathrm{1}} \left({t}\right)=\mathrm{tan}\:{t} \\ $$$${f}_{\mathrm{2}} \left({t}\right)=\frac{{t}\left({t}^{\mathrm{4}} +\mathrm{2}{t}^{\mathrm{2}} +\mathrm{1}\right)}{\mathrm{3}{t}^{\mathrm{4}} −\mathrm{4}{t}^{\mathrm{2}} +\mathrm{1}}=\frac{{t}}{\mathrm{3}}+\frac{\mathrm{2}{t}\left(\mathrm{5}{t}^{\mathrm{2}} +\mathrm{1}\right)}{\mathrm{3}\left(\mathrm{3}{t}^{\mathrm{4}} −\mathrm{4}{t}^{\mathrm{2}} +\mathrm{1}\right)} \\ $$$$\Rightarrow\:{g}\left({t}\right)=\frac{\mathrm{1}}{\mathrm{3}}{t}\:\mathrm{is}\:\mathrm{asymptote}\:\mathrm{of}\:{f}_{\mathrm{1}} \left({t}\right) \\ $$$$\mathrm{and}\:\mathrm{obviously}\:\mathrm{tan}\:{t}\:={at}\:\mathrm{with}\:{a}\in\mathbb{R}\:\mathrm{has} \\ $$$$\mathrm{infinite}\:\mathrm{solutions} \\ $$

Question Number 118369 Answers: 0 Comments: 1

657×10^4 =(5/7)+

$$\:\mathrm{657}×\mathrm{10}^{\mathrm{4}} =\frac{\mathrm{5}}{\mathrm{7}}+ \\ $$

Question Number 118366 Answers: 0 Comments: 4

4. Turunan fungsi f(x)=^5 (√((10x^2 −4)^8 )) adalah f^1 (x). Nilai f^1 (1)=... a. 1 b. 8 c. 14 d. 16 e. 20

$$\mathrm{4}.\:\mathrm{Turunan}\:\mathrm{fungsi}\:\mathrm{f}\left(\mathrm{x}\right)=^{\mathrm{5}} \sqrt{\left(\mathrm{10x}^{\mathrm{2}} −\mathrm{4}\right)^{\mathrm{8}} \:\:} \\ $$$$\mathrm{adalah}\:\mathrm{f}^{\mathrm{1}} \left(\mathrm{x}\right).\:\mathrm{Nilai}\:\mathrm{f}^{\mathrm{1}} \left(\mathrm{1}\right)=... \\ $$$${a}.\:\mathrm{1} \\ $$$${b}.\:\mathrm{8} \\ $$$${c}.\:\mathrm{14} \\ $$$$\mathrm{d}.\:\mathrm{16} \\ $$$$\mathrm{e}.\:\mathrm{20} \\ $$

Question Number 118364 Answers: 1 Comments: 0

...ordinary differential equation... y′′ −xy′+y=0 general solution ::=?? .m.n.1970.

$$\:\:\:\:\:\:\:\:\:\:\:\:\:...{ordinary}\:{differential}\:{equation}... \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:{y}''\:−{xy}'+{y}=\mathrm{0} \\ $$$$\:\:\:\:{general}\:{solution}\:::=?? \\ $$$$\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:.{m}.{n}.\mathrm{1970}. \\ $$$$ \\ $$

Question Number 118357 Answers: 2 Comments: 0

Find the sum of the prime factors of 20^5 +21.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{prime}\:\mathrm{factors}\:\mathrm{of} \\ $$$$\mathrm{20}^{\mathrm{5}} +\mathrm{21}. \\ $$

Question Number 118346 Answers: 0 Comments: 0

Let 30 furniture sets arrive at two city stations A and B ,15 sets for each station All funiture sets need to be delivered to two furniture stores C and D,and 10 sets must be delivered to store C,and to store D−20.It is known that delivery one furniture set from the station A to the stores C and D costs 1 and 3 monetary units,and from station B−2 and 5 units respectively,.It is necessary to draw out such a transportation plan that the cost of transportation is the lowest

$$\mathrm{Let}\:\mathrm{30}\:\mathrm{furniture}\:\mathrm{sets}\:\mathrm{arrive}\:\mathrm{at}\:\mathrm{two}\:\mathrm{city} \\ $$$$\mathrm{stations}\:\mathrm{A}\:\mathrm{and}\:\mathrm{B}\:,\mathrm{15}\:\mathrm{sets}\:\mathrm{for}\:\mathrm{each}\:\mathrm{station} \\ $$$$\mathrm{All}\:\mathrm{funiture}\:\mathrm{sets}\:\mathrm{need}\:\mathrm{to}\:\mathrm{be}\:\mathrm{delivered} \\ $$$$\mathrm{to}\:\mathrm{two}\:\mathrm{furniture}\:\mathrm{stores}\:\mathrm{C}\:\mathrm{and}\:\mathrm{D},\mathrm{and}\:\mathrm{10} \\ $$$$\mathrm{sets}\:\mathrm{must}\:\mathrm{be}\:\mathrm{delivered}\:\mathrm{to}\:\mathrm{store}\:\mathrm{C},\mathrm{and} \\ $$$$\mathrm{to}\:\mathrm{store}\:\mathrm{D}−\mathrm{20}.\mathrm{It}\:\mathrm{is}\:\mathrm{known}\:\mathrm{that}\:\mathrm{delivery} \\ $$$$\mathrm{one}\:\mathrm{furniture}\:\mathrm{set}\:\mathrm{from}\:\mathrm{the}\:\mathrm{station}\:\mathrm{A}\:\mathrm{to} \\ $$$$\mathrm{the}\:\mathrm{stores}\:\mathrm{C}\:\mathrm{and}\:\mathrm{D}\:\mathrm{costs}\:\mathrm{1}\:\mathrm{and}\:\mathrm{3}\:\mathrm{monetary} \\ $$$$\mathrm{units},\mathrm{and}\:\mathrm{from}\:\mathrm{station}\:\mathrm{B}−\mathrm{2}\:\mathrm{and}\:\mathrm{5}\:\mathrm{units} \\ $$$$\:\mathrm{respectively},.\mathrm{It}\:\mathrm{is}\:\mathrm{necessary}\:\mathrm{to}\:\mathrm{draw} \\ $$$$\mathrm{out}\:\mathrm{such}\:\mathrm{a}\:\mathrm{transportation}\:\mathrm{plan}\:\mathrm{that} \\ $$$$\mathrm{the}\:\mathrm{cost}\:\mathrm{of}\:\mathrm{transportation}\:\mathrm{is}\:\mathrm{the}\:\mathrm{lowest} \\ $$

Question Number 118349 Answers: 1 Comments: 0

solution xdy + (3y − e^x )dx with integration factor

$${solution}\:{xdy}\:+\:\left(\mathrm{3}{y}\:−\:{e}^{{x}} \right){dx}\: \\ $$$${with}\:{integration}\:{factor} \\ $$

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