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Question Number 67349 Answers: 1 Comments: 0

Solve for y(x) xy′ = y + 2x^3 sin^2 ((y/x))

$$\mathrm{Solve}\:\mathrm{for}\:{y}\left({x}\right) \\ $$$${xy}'\:=\:{y}\:+\:\mathrm{2}{x}^{\mathrm{3}} \mathrm{sin}^{\mathrm{2}} \left(\frac{{y}}{{x}}\right) \\ $$

Question Number 67345 Answers: 1 Comments: 0

3sinx+5cosx=5 then prove that 5sinx−3cox= +3

$$\mathrm{3}{sinx}+\mathrm{5}{cosx}=\mathrm{5}\:{then}\:{prove}\:{that}\: \\ $$$$\mathrm{5}{sinx}−\mathrm{3}{cox}=\:+\mathrm{3} \\ $$

Question Number 67342 Answers: 0 Comments: 1

find the value of ∫_(−∞) ^∞ ((sin(2x^2 ))/((x^2 −x +3)^3 ))dx

$${find}\:{the}\:{value}\:{of}\:\int_{−\infty} ^{\infty} \:\:\:\frac{{sin}\left(\mathrm{2}{x}^{\mathrm{2}} \right)}{\left({x}^{\mathrm{2}} −{x}\:+\mathrm{3}\right)^{\mathrm{3}} }{dx} \\ $$

Question Number 67310 Answers: 1 Comments: 2

calculate ∫_(−∞) ^(+∞) (dx/(x^4 +x^2 +1))

$${calculate}\:\int_{−\infty} ^{+\infty} \:\:\frac{{dx}}{{x}^{\mathrm{4}} \:+{x}^{\mathrm{2}} \:+\mathrm{1}} \\ $$

Question Number 67307 Answers: 1 Comments: 1

Question Number 67299 Answers: 2 Comments: 5

G(x)= (x+1)(x+3)Q(x) + px +q a) Given that G(x) leaves a remainder of 8 and −24 when divided by (x+1) and (x+3) respectively,find the remainder when G(x) is divided by (x+1)(x+3). b) Given that x+2 is a factor of G(x) and that the graph of G(x) passes through the point with coordinates (0,6) find G(x)

$${G}\left({x}\right)=\:\left({x}+\mathrm{1}\right)\left({x}+\mathrm{3}\right){Q}\left({x}\right)\:+\:{px}\:+{q} \\ $$$$\left.{a}\right)\:{Given}\:{that}\:{G}\left({x}\right)\:{leaves}\:{a}\:{remainder}\:{of}\:\mathrm{8}\:{and}\:−\mathrm{24}\:{when}\:{divided}\:{by}\:\left({x}+\mathrm{1}\right)\:{and}\: \\ $$$$\left({x}+\mathrm{3}\right)\:{respectively},{find}\:{the}\:{remainder}\:{when}\:{G}\left({x}\right)\:{is}\:{divided}\:{by}\:\left({x}+\mathrm{1}\right)\left({x}+\mathrm{3}\right). \\ $$$$\left.{b}\right)\:\:{Given}\:{that}\:{x}+\mathrm{2}\:{is}\:{a}\:{factor}\:{of}\:{G}\left({x}\right)\:{and}\:{that}\:{the}\:{graph}\:{of}\:{G}\left({x}\right)\:{passes}\:{through} \\ $$$${the}\:{point}\:{with}\:{coordinates}\:\left(\mathrm{0},\mathrm{6}\right)\:{find}\:{G}\left({x}\right) \\ $$

Question Number 67333 Answers: 0 Comments: 0

evaluate Σ_(n=0) ^(+∞) (1/((1+8n)^2 ))

$${evaluate}\:\underset{{n}=\mathrm{0}} {\overset{+\infty} {\sum}}\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{8}{n}\right)^{\mathrm{2}} } \\ $$

Question Number 67281 Answers: 1 Comments: 6

Question Number 67337 Answers: 1 Comments: 0

m(p+1)=a ((q(p+1))/(r+1))=b & ((p(p+1))/(q+1))=c and ((r(p+1))/(m+1))=d find either of p,q,r,m in terms of a,b,c,d.

$${m}\left({p}+\mathrm{1}\right)={a} \\ $$$$\frac{{q}\left({p}+\mathrm{1}\right)}{{r}+\mathrm{1}}={b}\:\:\&\:\:\frac{{p}\left({p}+\mathrm{1}\right)}{{q}+\mathrm{1}}={c} \\ $$$${and}\:\:\frac{{r}\left({p}+\mathrm{1}\right)}{{m}+\mathrm{1}}={d} \\ $$$${find}\:{either}\:{of}\:{p},{q},{r},{m}\:{in}\:{terms}\:{of} \\ $$$${a},{b},{c},{d}. \\ $$

Question Number 67336 Answers: 0 Comments: 3

If (dy/dx) = e^(−t) (dy/dt) , find (d^2 y/dx^2 )

$$\mathrm{If}\:\:\frac{\mathrm{dy}}{\mathrm{dx}}\:=\:\mathrm{e}^{−\mathrm{t}} \:\:\frac{\mathrm{dy}}{\mathrm{dt}}\:\:,\:\:\:\:\:\:\mathrm{find}\:\:\:\frac{\mathrm{d}^{\mathrm{2}} \mathrm{y}}{\mathrm{dx}^{\mathrm{2}} } \\ $$

Question Number 67254 Answers: 1 Comments: 10

Question Number 67246 Answers: 2 Comments: 4

Integrate: 1) ∫_3 ^( ∞) ((1/x dx)/(ln(x)(√(ln^2 x−1)))) 2) ∫_1 ^∞ ((e^x dx)/(1+e^(2x) )) 3) ∫_1 ^∞ ((2^x dx)/(x+1)) 4) ∫_2 ^∞ ((√x)/(ln(x)))dx

$${Integrate}: \\ $$$$\left.\mathrm{1}\right)\:\underset{\mathrm{3}} {\overset{\:\infty} {\int}}\:\frac{\mathrm{1}/{x}\:{dx}}{{ln}\left({x}\right)\sqrt{{ln}^{\mathrm{2}} {x}−\mathrm{1}}} \\ $$$$\left.\mathrm{2}\right)\:\underset{\mathrm{1}} {\overset{\infty} {\int}}\frac{{e}^{{x}} {dx}}{\mathrm{1}+{e}^{\mathrm{2}{x}} } \\ $$$$\left.\mathrm{3}\right)\:\underset{\mathrm{1}} {\overset{\infty} {\int}}\:\frac{\mathrm{2}^{{x}} {dx}}{{x}+\mathrm{1}} \\ $$$$\left.\mathrm{4}\right)\:\underset{\mathrm{2}} {\overset{\infty} {\int}}\:\frac{\sqrt{{x}}}{{ln}\left({x}\right)}{dx} \\ $$

Question Number 67244 Answers: 0 Comments: 7

Which of the series converge and which diverge? Check by the limit comparison test. 1) Σ_(n=2) ^∞ ((1+n ln(n))/(n^2 +5)) 2) Σ_(n=1) ^∞ ((ln(n))/n^(3/2) ) 3) Σ_(n=3) ^∞ (1/(ln(lnn))) 4) Σ_(n=1) ^∞ (1/(n (n)^(1/n) )) ??

$${Which}\:{of}\:{the}\:{series}\:{converge}\:{and}\: \\ $$$${which}\:{diverge}?\:{Check}\:{by}\:{the}\:{limit} \\ $$$${comparison}\:{test}. \\ $$$$\left.\mathrm{1}\right)\:\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}+{n}\:{ln}\left({n}\right)}{{n}^{\mathrm{2}} +\mathrm{5}} \\ $$$$\left.\mathrm{2}\right)\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{{ln}\left({n}\right)}{{n}^{\frac{\mathrm{3}}{\mathrm{2}}} } \\ $$$$\left.\mathrm{3}\right)\:\underset{{n}=\mathrm{3}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{{ln}\left({lnn}\right)} \\ $$$$\left.\mathrm{4}\right)\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{{n}\:\left({n}\right)^{\frac{\mathrm{1}}{{n}}} }\:\: \\ $$$$?? \\ $$

Question Number 67236 Answers: 1 Comments: 1

let T_n =cos(narccosx) 1) calculste T_0 ,T_1 ,T_2 2)find roots of T_n 3)decompose the fraction F =(1/T_n )

$${let}\:{T}_{{n}} ={cos}\left({narccosx}\right) \\ $$$$\left.\mathrm{1}\right)\:{calculste}\:{T}_{\mathrm{0}} ,{T}_{\mathrm{1}} ,{T}_{\mathrm{2}} \\ $$$$\left.\mathrm{2}\right){find}\:\:{roots}\:{of}\:{T}_{{n}} \\ $$$$\left.\mathrm{3}\right){decompose}\:\:{the}\:{fraction}\:{F}\:=\frac{\mathrm{1}}{{T}_{{n}} } \\ $$

Question Number 67235 Answers: 0 Comments: 1

find ∫_(−(π/3)) ^(π/3) x^2 {cosx−sinx}^3 dx

$${find}\:\:\int_{−\frac{\pi}{\mathrm{3}}} ^{\frac{\pi}{\mathrm{3}}} \:{x}^{\mathrm{2}} \left\{{cosx}−{sinx}\right\}^{\mathrm{3}} {dx} \\ $$

Question Number 67234 Answers: 2 Comments: 3

factorise p(x)=1+x+x^2 +x^3 +x^5 inside C[x] and R[x] calculate p(e^(i(π/5)) ) and p(cos((π/5)))

$${factorise}\:{p}\left({x}\right)=\mathrm{1}+{x}+{x}^{\mathrm{2}} \:+{x}^{\mathrm{3}} \:+{x}^{\mathrm{5}} \\ $$$${inside}\:{C}\left[{x}\right]\:{and}\:{R}\left[{x}\right] \\ $$$${calculate}\:{p}\left({e}^{{i}\frac{\pi}{\mathrm{5}}} \right)\:{and}\:{p}\left({cos}\left(\frac{\pi}{\mathrm{5}}\right)\right) \\ $$

Question Number 67233 Answers: 0 Comments: 1

calculate ∫_0 ^1 ((xdx)/(√(1+x^4 )))

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{xdx}}{\sqrt{\mathrm{1}+{x}^{\mathrm{4}} }} \\ $$

Question Number 67232 Answers: 1 Comments: 1

calculate Σ_(n=1) ^∞ ((cos(n(π/3)))/n)

$${calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{{cos}\left({n}\frac{\pi}{\mathrm{3}}\right)}{{n}} \\ $$

Question Number 67231 Answers: 2 Comments: 0

find ∫x/x^5 −1) dx

$$\left.{find}\:\int{x}/{x}^{\mathrm{5}} −\mathrm{1}\right)\:{dx} \\ $$

Question Number 67298 Answers: 0 Comments: 2

Find the third degree polynomial which vanishes when x =−1 and x = 2, which has a value 8 when x =0 and leaves a remainder ((16)/3) when divided by 3x + 2.

$${Find}\:\:{the}\:{third}\:{degree}\:{polynomial}\:{which}\:{vanishes}\:{when} \\ $$$${x}\:=−\mathrm{1}\:{and}\:{x}\:=\:\mathrm{2},\:{which}\:{has}\:{a}\:{value}\:\mathrm{8}\:{when}\:{x}\:=\mathrm{0}\:{and}\:{leaves}\:{a}\:{remainder}\:\frac{\mathrm{16}}{\mathrm{3}}\:{when} \\ $$$${divided}\:{by}\:\:\mathrm{3}{x}\:+\:\mathrm{2}. \\ $$

Question Number 67294 Answers: 0 Comments: 3

solve for x and y the simultaneous equation log_3 x = y = log(2x − 1)

$${solve}\:{for}\:\:\mathrm{x}\:\mathrm{and}\:\mathrm{y}\:{the}\:{simultaneous}\:{equation} \\ $$$$\:\mathrm{log}_{\mathrm{3}} {x}\:=\:{y}\:=\:\mathrm{log}\left(\mathrm{2}{x}\:−\:\mathrm{1}\right) \\ $$

Question Number 67215 Answers: 1 Comments: 1

Question Number 67208 Answers: 1 Comments: 6

Find the times in a day when the hour′s, minute′s and second′s hand of a clock occupy the same angular position. [old question reposted]

$${Find}\:{the}\:{times}\:{in}\:{a}\:{day}\:{when} \\ $$$${the}\:{hour}'{s},\:{minute}'{s}\:{and}\:{second}'{s} \\ $$$${hand}\:{of}\:{a}\:{clock}\:{occupy}\:{the}\:{same} \\ $$$${angular}\:{position}. \\ $$$$\left[{old}\:{question}\:{reposted}\right] \\ $$

Question Number 67197 Answers: 1 Comments: 0

∫_0 ^2 x^5 (1−(x/2))^4 dx

$$\int_{\mathrm{0}} ^{\mathrm{2}} {x}^{\mathrm{5}} \left(\mathrm{1}−\frac{{x}}{\mathrm{2}}\right)^{\mathrm{4}} {dx} \\ $$

Question Number 67193 Answers: 0 Comments: 7

Question Number 67189 Answers: 1 Comments: 0

solve inside R^3 the system { ((2x+y+z =1)),((x+2y+z =2)) :} {x+y+2z =3

$${solve}\:{inside}\:{R}^{\mathrm{3}} \:{the}\:{system}\:\begin{cases}{\mathrm{2}{x}+{y}+{z}\:=\mathrm{1}}\\{{x}+\mathrm{2}{y}+{z}\:=\mathrm{2}}\end{cases} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left\{{x}+{y}+\mathrm{2}{z}\:=\mathrm{3}\right. \\ $$

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