Question and Answers Forum

let j =e^((i2π)/3) and P(x) =(1+jx)^n −(1−jx)^n 1) find P(x) at form of arctan 2) find the roots of P(x) 3)factorize inside C[x] the polynome P(x) 4) calculate ∫_0 ^1 P(x)dx

$${let}\:{j}\:={e}^{\frac{{i}\mathrm{2}\pi}{\mathrm{3}}} \:\:\:{and}\:{P}\left({x}\right)\:=\left(\mathrm{1}+{jx}\right)^{{n}} −\left(\mathrm{1}−{jx}\right)^{{n}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{P}\left({x}\right)\:{at}\:{form}\:{of}\:{arctan} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{roots}\:{of}\:{P}\left({x}\right) \\ $$$$\left.\mathrm{3}\right){factorize}\:{inside}\:{C}\left[{x}\right]\:\:{the}\:{polynome}\:{P}\left({x}\right) \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{P}\left({x}\right){dx} \\ $$

Question Number 62241 Answers: 1 Comments: 0

if the point A B C with position vector (20i^ +λj^ ) (5i^ −j^ ) and(10i^ −13j^ ) are collinear then the value of λ is:

$$\boldsymbol{{if}}\:\boldsymbol{{the}}\:\boldsymbol{{point}}\:{A}\:{B}\:{C}\:{with}\:{position}\:{vector}\: \\ $$$$\left(\mathrm{20}\hat {{i}}+\lambda\hat {{j}}\right)\:\left(\mathrm{5}\hat {{i}}−\hat {{j}}\right)\:{and}\left(\mathrm{10}\hat {{i}}−\mathrm{13}\hat {{j}}\right)\:{are} \\ $$$${collinear}\:{then}\:{the}\:{value}\:{of}\:\lambda\:{is}: \\ $$

Question Number 62220 Answers: 0 Comments: 2

let f(x) =∫_0 ^∞ (t^2 /(x^6 +t^6 )) dt with x>0 1) calculate f(x) 2) calculate g(x) =∫_0 ^∞ (t^2 /((x^6 +t^6 )^2 ))dt 3) find values of integrals ∫_0 ^∞ (t^2 /(t^6 +8))dt and ∫_0 ^∞ (t^2 /((t^6 +8)^2 ))dt .

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\:\:\frac{{t}^{\mathrm{2}} }{{x}^{\mathrm{6}} \:\:+{t}^{\mathrm{6}} }\:{dt}\:\:\:\:\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{t}^{\mathrm{2}} }{\left({x}^{\mathrm{6}} \:+{t}^{\mathrm{6}} \right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{values}\:{of}\:{integrals}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{t}^{\mathrm{2}} }{{t}^{\mathrm{6}} \:+\mathrm{8}}{dt}\:\:\:\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{t}^{\mathrm{2}} }{\left({t}^{\mathrm{6}} +\mathrm{8}\right)^{\mathrm{2}} }{dt}\:. \\ $$

Question Number 62214 Answers: 1 Comments: 0

Find out x,y such that ((lcm(x,y))/(gcd(x,y)))=lcm(x,y)−gcd(x,y)

$$\mathrm{Find}\:\mathrm{out}\:\mathrm{x},\mathrm{y}\:\mathrm{such}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{lcm}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{gcd}\left(\mathrm{x},\mathrm{y}\right)}=\mathrm{lcm}\left(\mathrm{x},\mathrm{y}\right)−\mathrm{gcd}\left(\mathrm{x},\mathrm{y}\right) \\ $$

Question Number 62213 Answers: 0 Comments: 1

calculate ∫∫∫_D e^(−x^2 −y^2 ) (√(x^2 +y^2 +z^2 ))dxdydz with D ={(x,y,z)∈R^3 / 0≤x≤1 , 1≤y≤2 and 2≤z≤3 }

$${calculate}\:\int\int\int_{{D}} \:{e}^{−{x}^{\mathrm{2}} −{y}^{\mathrm{2}} } \sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:+{z}^{\mathrm{2}} }{dxdydz}\:{with} \\ $$$${D}\:=\left\{\left({x},{y},{z}\right)\in{R}^{\mathrm{3}} \:/\:\:\:\mathrm{0}\leqslant{x}\leqslant\mathrm{1}\:,\:\mathrm{1}\leqslant{y}\leqslant\mathrm{2}\:\:{and}\:\:\:\mathrm{2}\leqslant{z}\leqslant\mathrm{3}\:\right\} \\ $$

Question Number 62211 Answers: 0 Comments: 3

(x/((√(4−x^2 ))+3))Max=(5/3)?

$$\frac{{x}}{\sqrt{\mathrm{4}−{x}^{\mathrm{2}} }+\mathrm{3}}{Max}=\frac{\mathrm{5}}{\mathrm{3}}? \\ $$

Question Number 62210 Answers: 0 Comments: 2

let f(x) =(x+1)^n arctan(nx) calculate f^((n)) (0).