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Question Number 42260 Answers: 0 Comments: 2

let f(a) = ∫_(−∞) ^(+∞) cos(ax^2 )dx with a>0 1) calculate f(a) interms of a ) calculate ∫_(−∞) ^(+∞) cos(2x^2 )dx 3) find the value of ∫_(−∞) ^(+∞) cos(x^2 +x+1)dx .

$${let}\:{f}\left({a}\right)\:=\:\int_{−\infty} ^{+\infty} \:{cos}\left({ax}^{\mathrm{2}} \right){dx}\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({a}\right)\:{interms}\:{of}\:{a} \\ $$$$\left.\right)\:{calculate}\:\int_{−\infty} ^{+\infty} \:\:\:{cos}\left(\mathrm{2}{x}^{\mathrm{2}} \right){dx} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{−\infty} ^{+\infty} \:{cos}\left({x}^{\mathrm{2}} \:+{x}+\mathrm{1}\right){dx}\:. \\ $$

Question Number 42305 Answers: 0 Comments: 6

let f(x) = ∫_(−∞) ^(+∞) ((cos(xt))/((t−i)^2 )) dt 1) let R =Re(f(x)) and I =Im(f(x)) extract R and I 2) calculate R and I 3) conclude the value of f(x) 4) calculate ∫_(−∞) ^(+∞) ((cos(2t))/((t−i)^2 ))dt 5) let u_n = ∫_(−∞) ^(+∞) ((cos((t/n)))/((t−i)^2 ))dt (n natral integer not o) find lim_(n→+∞) u_n and study the convergence of Σu_n

$${let}\:{f}\left({x}\right)\:=\:\int_{−\infty} ^{+\infty} \:\:\frac{{cos}\left({xt}\right)}{\left({t}−{i}\right)^{\mathrm{2}} }\:{dt} \\ $$$$\left.\mathrm{1}\right)\:{let}\:{R}\:={Re}\left({f}\left({x}\right)\right)\:{and}\:{I}\:={Im}\left({f}\left({x}\right)\right)\:{extract}\:{R}\:{and}\:{I} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{R}\:{and}\:{I} \\ $$$$\left.\mathrm{3}\right)\:{conclude}\:{the}\:{value}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{cos}\left(\mathrm{2}{t}\right)}{\left({t}−{i}\right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{5}\right)\:{let}\:{u}_{{n}} =\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{cos}\left(\frac{{t}}{{n}}\right)}{\left({t}−{i}\right)^{\mathrm{2}} }{dt}\:\:\:\:\left({n}\:{natral}\:{integer}\:{not}\:{o}\right) \\ $$$${find}\:{lim}_{{n}\rightarrow+\infty} {u}_{{n}} \:\:\:\:{and}\:\:{study}\:{the}\:{convergence}\:{of}\:\Sigma{u}_{{n}} \\ $$

Question Number 42228 Answers: 1 Comments: 0

Solve : 2x^2 ydx −2y^4 dx+2x^3 dy+3xy^3 dy=0.

$$\mathrm{Solve}\:: \\ $$$$\mathrm{2}{x}^{\mathrm{2}} {ydx}\:−\mathrm{2}{y}^{\mathrm{4}} {dx}+\mathrm{2}{x}^{\mathrm{3}} {dy}+\mathrm{3}{xy}^{\mathrm{3}} {dy}=\mathrm{0}. \\ $$

Question Number 42224 Answers: 0 Comments: 5

a + b + c = 180 a,b,c ∈ N number of triplets possible (a,b,c) for the above equation are ? ( the order of a,b,c doesn′t matter)

$${a}\:+\:{b}\:+\:{c}\:=\:\mathrm{180}\: \\ $$$${a},{b},{c}\:\in\:\mathbb{N} \\ $$$${number}\:{of}\:{triplets}\:{possible} \\ $$$$\left({a},{b},{c}\right)\:{for}\:{the}\:{above} \\ $$$${equation}\:{are}\:? \\ $$$$\left(\:{the}\:{order}\:{of}\:{a},{b},{c}\:\:{doesn}'{t}\right. \\ $$$$\left.{matter}\right) \\ $$

Question Number 42222 Answers: 1 Comments: 0

let f(x) =e^(−∣x∣) , 2π periodic even developp f at fourier serie .

$${let}\:{f}\left({x}\right)\:={e}^{−\mid{x}\mid} \:,\:\:\mathrm{2}\pi\:{periodic}\:{even}\:\:{developp}\:{f}\:{at}\:{fourier}\:{serie}\:. \\ $$

Question Number 42221 Answers: 1 Comments: 0

Solve: (dt/dx) = (2/(x+t)) .

$$\mathrm{Solve}: \\ $$$$\frac{\mathrm{dt}}{\mathrm{d}{x}}\:=\:\frac{\mathrm{2}}{{x}+\mathrm{t}}\:. \\ $$

Question Number 42215 Answers: 0 Comments: 3

Number of straight lines which satisfy the differential equation (dy/dx) + x((dy/dx))^2 − y =0 is ?

$$\mathrm{Number}\:\mathrm{of}\:\mathrm{straight}\:\mathrm{lines}\:\mathrm{which}\:\mathrm{satisfy} \\ $$$$\mathrm{the}\:\mathrm{differential}\:\mathrm{equation} \\ $$$$\frac{\mathrm{dy}}{{dx}}\:+\:{x}\left(\frac{{dy}}{{dx}}\right)^{\mathrm{2}} −\:{y}\:=\mathrm{0}\:{is}\:? \\ $$

Question Number 42207 Answers: 1 Comments: 0

Find the equation of tangent and normal to the curve y given by y = x^3 + 3x^2 + 7 .

$${Find}\:{the}\:{equation}\:{of}\:{tangent}\:{and}\:{normal}\:{to}\:\:{the}\:{curve}\:{y} \\ $$$${given}\:{by}\:\:\:{y}\:=\:{x}^{\mathrm{3}} \:+\:\mathrm{3}{x}^{\mathrm{2}} \:+\:\mathrm{7}\:. \\ $$

Question Number 42200 Answers: 1 Comments: 0

Question Number 42199 Answers: 1 Comments: 0

Question Number 42196 Answers: 0 Comments: 4

Let P be an interior point of a triangle ABC and AP,BP,CP meet the sides BC, CA,AB in D,E,F respectively. Show that ((AP)/(PD))= ((AF)/(FB)) + ((AE)/(EC)) .

$$\mathrm{Let}\:\mathrm{P}\:\mathrm{be}\:\mathrm{an}\:\mathrm{interior}\:\mathrm{point}\:\mathrm{of}\:\mathrm{a}\:\mathrm{triangle} \\ $$$$\mathrm{ABC}\:\mathrm{and}\:\mathrm{AP},\mathrm{BP},\mathrm{CP}\:\mathrm{meet}\:\mathrm{the}\:\mathrm{sides}\:\mathrm{BC}, \\ $$$$\mathrm{CA},\mathrm{AB}\:\mathrm{in}\:\mathrm{D},\mathrm{E},\mathrm{F}\:\mathrm{respectively}.\:\mathrm{Show} \\ $$$$\mathrm{that}\:\frac{\mathrm{AP}}{\mathrm{PD}}=\:\frac{\mathrm{AF}}{\mathrm{FB}}\:+\:\frac{\mathrm{AE}}{\mathrm{EC}}\:. \\ $$

Question Number 42195 Answers: 1 Comments: 0

let p(x)=x^(10) −1 1) find roots of p(x) 2) factorize i nside C[x] p(x{ 3) factorize inside R[x] p(x) .

$${let}\:{p}\left({x}\right)={x}^{\mathrm{10}} −\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{roots}\:{of}\:{p}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{factorize}\:{i}\:{nside}\:\:{C}\left[{x}\right]\:{p}\left({x}\left\{\right.\right. \\ $$$$\left.\mathrm{3}\right)\:{factorize}\:{inside}\:{R}\left[{x}\right]\:{p}\left({x}\right)\:. \\ $$

Question Number 42410 Answers: 1 Comments: 1

∫_( 0) ^(2a) ((f(x))/(f(x)+f(2a−x))) dx =

$$\underset{\:\mathrm{0}} {\overset{\mathrm{2}{a}} {\int}}\:\:\frac{{f}\left({x}\right)}{{f}\left({x}\right)+{f}\left(\mathrm{2}{a}−{x}\right)}\:{dx}\:= \\ $$

Question Number 42191 Answers: 0 Comments: 1

let A_p =∫_0 ^∞ ((sin(px))/(e^x −1)) dx with p>0 1)give A_p at form of serie 2) give A_1 at form of serie .

$${let}\:{A}_{{p}} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sin}\left({px}\right)}{{e}^{{x}} −\mathrm{1}}\:{dx}\:\:{with}\:{p}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right){give}\:{A}_{{p}} \:\:{at}\:{form}\:{of}\:{serie} \\ $$$$\left.\mathrm{2}\right)\:{give}\:{A}_{\mathrm{1}} \:{at}\:{form}\:{of}\:{serie}\:. \\ $$

Question Number 42190 Answers: 0 Comments: 1

let u_n =Σ_(k=1) ^n (((−1)^k )/(√k)) 1) prove that (u_n )is convergente 2) find a equivalent of u_n when n→+∞