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Relation and FunctionsQuestion and Answers: Page 7

Question Number 147972 Answers: 0 Comments: 0

find ∫_0 ^∞ ((arctan(2x+1))/(x^2 +4))dx

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{arctan}\left(\mathrm{2x}+\mathrm{1}\right)}{\mathrm{x}^{\mathrm{2}} \:+\mathrm{4}}\mathrm{dx} \\ $$

Question Number 147971 Answers: 0 Comments: 6

find ∫_(−∞) ^(+∞) ((x^3 dx)/((x^2 +x+1)^4 ))

$$\mathrm{find}\:\int_{−\infty} ^{+\infty} \:\frac{\mathrm{x}^{\mathrm{3}} \mathrm{dx}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}\right)^{\mathrm{4}} } \\ $$

Question Number 147863 Answers: 1 Comments: 0

calculate Σ_(n=0) ^∞ ((n!^2 )/((2n)!))

$$\mathrm{calculate}\:\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \:\frac{\mathrm{n}!^{\mathrm{2}} }{\left(\mathrm{2n}\right)!} \\ $$$$ \\ $$

Question Number 147861 Answers: 0 Comments: 0

resoudre dans Z^2 x^2 −y^2 =3x

$$\mathrm{resoudre}\:\mathrm{dans}\:\mathrm{Z}^{\mathrm{2}} \:\:\:\:\mathrm{x}^{\mathrm{2}} −\mathrm{y}^{\mathrm{2}} \:=\mathrm{3x} \\ $$

Question Number 147688 Answers: 1 Comments: 0

find lim_(n→+∞) ∫_(1/n) ^(√n) xe^(−x^2 ) arctan(nx)dx

$$\mathrm{find}\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \int_{\frac{\mathrm{1}}{\mathrm{n}}} ^{\sqrt{\mathrm{n}}} \:\:\:\mathrm{xe}^{−\mathrm{x}^{\mathrm{2}} } \mathrm{arctan}\left(\mathrm{nx}\right)\mathrm{dx} \\ $$

Question Number 147687 Answers: 2 Comments: 0

calculate lim_(n→0) ((e^(−nx^2 ) −nx−1)/x^3 )

$$\mathrm{calculate}\:\mathrm{lim}_{\mathrm{n}\rightarrow\mathrm{0}} \:\frac{\mathrm{e}^{−\mathrm{nx}^{\mathrm{2}} } −\mathrm{nx}−\mathrm{1}}{\mathrm{x}^{\mathrm{3}} } \\ $$

Question Number 147685 Answers: 2 Comments: 0

calculate lim_(x→0) ((sin(2sinx))/x^2 )

$$\mathrm{calculate}\:\mathrm{lim}_{\mathrm{x}\rightarrow\mathrm{0}} \:\frac{\mathrm{sin}\left(\mathrm{2sinx}\right)}{\mathrm{x}^{\mathrm{2}} } \\ $$

Question Number 147684 Answers: 0 Comments: 0

decompse F(x)=(x^3 /((x^2 +1)^4 )) inside C(x)

$$\mathrm{decompse}\:\mathrm{F}\left(\mathrm{x}\right)=\frac{\mathrm{x}^{\mathrm{3}} }{\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{4}} }\:\:\:\mathrm{inside}\:\mathrm{C}\left(\mathrm{x}\right) \\ $$

Question Number 147678 Answers: 0 Comments: 0

roots of Υ_n (x)=sin(narcsinx) (n integr natural) deompose F(x)=(1/(Υ_n (x)))

$$\mathrm{roots}\:\mathrm{of}\:\:\Upsilon_{\mathrm{n}} \left(\mathrm{x}\right)=\mathrm{sin}\left(\mathrm{narcsinx}\right)\:\:\left(\mathrm{n}\:\mathrm{integr}\:\mathrm{natural}\right) \\ $$$$\mathrm{deompose}\:\mathrm{F}\left(\mathrm{x}\right)=\frac{\mathrm{1}}{\Upsilon_{\mathrm{n}} \left(\mathrm{x}\right)} \\ $$

Question Number 147543 Answers: 2 Comments: 0

Π_(m=1) ^n ((1/2))^m

$$\underset{{m}=\mathrm{1}} {\overset{{n}} {\prod}}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{{m}} \\ $$

Question Number 147469 Answers: 0 Comments: 1

Question Number 147467 Answers: 3 Comments: 0

f(x)=x^n e^(−x) 1) calculate f^((n)) (0) and f^((n)) (1) 2)developp f at integr serie 3) calculate ∫_0 ^1 f(x)dx

$$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}^{\mathrm{n}} \:\mathrm{e}^{−\mathrm{x}} \\ $$$$\left.\mathrm{1}\right)\:\mathrm{calculate}\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{0}\right)\:\mathrm{and}\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{1}\right) \\ $$$$\left.\mathrm{2}\right)\mathrm{developp}\:\mathrm{f}\:\mathrm{at}\:\mathrm{integr}\:\mathrm{serie} \\ $$$$\left.\mathrm{3}\right)\:\mathrm{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} \\ $$

Question Number 147466 Answers: 2 Comments: 0

f(x)=x^2 −2x+5 find ∫ ((f(x))/(f^(−1) (x)))dx and ∫ ((f^(−1) (x))/(f(x)))dx

$$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}^{\mathrm{2}} −\mathrm{2x}+\mathrm{5} \\ $$$$\mathrm{find}\:\int\:\frac{\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{f}^{−\mathrm{1}} \left(\mathrm{x}\right)}\mathrm{dx}\:\:\:\mathrm{and}\:\int\:\:\frac{\mathrm{f}^{−\mathrm{1}} \left(\mathrm{x}\right)}{\mathrm{f}\left(\mathrm{x}\right)}\mathrm{dx} \\ $$

Question Number 147302 Answers: 1 Comments: 0

P_a (z)=z^(2n) −2z^n cosa+1 montrer que p_a (z)=Π_(k=0) ^(n−1) (z^2 −2zcos((a/π)+((2kπ)/n))+1)

$${P}_{{a}} \left({z}\right)={z}^{\mathrm{2}{n}} −\mathrm{2}{z}^{{n}} {cosa}+\mathrm{1} \\ $$$${montrer}\:{que}\:\:{p}_{{a}} \left(\mathrm{z}\right)=\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\prod}}\left({z}^{\mathrm{2}} −\mathrm{2}{zcos}\left(\frac{{a}}{\pi}+\frac{\mathrm{2}{k}\pi}{{n}}\right)+\mathrm{1}\right) \\ $$

Question Number 147258 Answers: 0 Comments: 0

Question Number 147218 Answers: 0 Comments: 0

calculate ∫_(∣z−1∣=2) (e^z /((z+i(√2))^2 (z+i)^2 (2z−1)))dz

$$\mathrm{calculate}\:\int_{\mid\mathrm{z}−\mathrm{1}\mid=\mathrm{2}} \:\:\:\:\frac{\mathrm{e}^{\mathrm{z}} }{\left(\mathrm{z}+\mathrm{i}\sqrt{\mathrm{2}}\right)^{\mathrm{2}} \left(\mathrm{z}+\mathrm{i}\right)^{\mathrm{2}} \left(\mathrm{2z}−\mathrm{1}\right)}\mathrm{dz} \\ $$

Question Number 147205 Answers: 1 Comments: 0

calculate ∫_1 ^∞ ((arctan((3/x)))/(2x^2 +1))dx

$$\mathrm{calculate}\:\int_{\mathrm{1}} ^{\infty} \:\frac{\mathrm{arctan}\left(\frac{\mathrm{3}}{\mathrm{x}}\right)}{\mathrm{2x}^{\mathrm{2}} \:+\mathrm{1}}\mathrm{dx} \\ $$

Question Number 147204 Answers: 0 Comments: 0

find ∫_0 ^1 lnxln(1−x)ln(1−x^2 )dx

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{lnxln}\left(\mathrm{1}−\mathrm{x}\right)\mathrm{ln}\left(\mathrm{1}−\mathrm{x}^{\mathrm{2}} \right)\mathrm{dx} \\ $$

Question Number 147201 Answers: 0 Comments: 0

f(x)=cos(sinx) developp f at fourier serie

$$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{cos}\left(\mathrm{sinx}\right)\:\:\mathrm{developp}\:\mathrm{f}\:\mathrm{at}\:\mathrm{fourier}\:\mathrm{serie} \\ $$

Question Number 147100 Answers: 0 Comments: 0

findA_n = ∫_0 ^1 x(x+1)(x+2)....(x+n)dx

$$\mathrm{findA}_{\mathrm{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{x}\left(\mathrm{x}+\mathrm{1}\right)\left(\mathrm{x}+\mathrm{2}\right)....\left(\mathrm{x}+\mathrm{n}\right)\mathrm{dx} \\ $$

Question Number 147006 Answers: 1 Comments: 0

find I_n =∫_0 ^∞ (dx/((x^2 +1)(x^2 +2)......(x^2 +n)))

$$\mathrm{find}\:\:\mathrm{I}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{\mathrm{dx}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{1}\right)\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{2}\right)......\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{n}\right)} \\ $$

Question Number 146902 Answers: 1 Comments: 0

let α and β roots of z^2 +3z+5=0 simlify U_n = Σ_(k=0) ^n (α^k +β^k ) and V_n =Σ_(k=0) ^n ((1/α^k )+(1/β^k ))

$$\mathrm{let}\:\alpha\:\mathrm{and}\:\beta\:\mathrm{roots}\:\mathrm{of}\:\:\mathrm{z}^{\mathrm{2}} +\mathrm{3z}+\mathrm{5}=\mathrm{0} \\ $$$$\mathrm{simlify}\:\mathrm{U}_{\mathrm{n}} =\:\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{n}} \:\left(\alpha^{\mathrm{k}} \:+\beta^{\mathrm{k}} \right) \\ $$$$\mathrm{and}\:\mathrm{V}_{\mathrm{n}} =\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{n}} \:\left(\frac{\mathrm{1}}{\alpha^{\mathrm{k}} }+\frac{\mathrm{1}}{\beta^{\mathrm{k}} }\right) \\ $$

Question Number 146901 Answers: 1 Comments: 0

g(x)=cos(2arcsinx) calculate (dg/dx) and (d^2 g/dx^2 ) 2)find ∫_(−(1/2)) ^(1/2) g(x)dx

$$\mathrm{g}\left(\mathrm{x}\right)=\mathrm{cos}\left(\mathrm{2arcsinx}\right)\:\: \\ $$$$\mathrm{calculate}\:\frac{\mathrm{dg}}{\mathrm{dx}}\:\mathrm{and}\:\frac{\mathrm{d}^{\mathrm{2}} \mathrm{g}}{\mathrm{dx}^{\mathrm{2}} } \\ $$$$\left.\mathrm{2}\right)\mathrm{find}\:\int_{−\frac{\mathrm{1}}{\mathrm{2}}} ^{\frac{\mathrm{1}}{\mathrm{2}}} \:\mathrm{g}\left(\mathrm{x}\right)\mathrm{dx} \\ $$

Question Number 146899 Answers: 1 Comments: 0

f(x)=sin^5 x calculate f^((5)) ((π/2))

$$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{sin}^{\mathrm{5}} \mathrm{x}\:\:\:\mathrm{calculate}\:\mathrm{f}^{\left(\mathrm{5}\right)} \left(\frac{\pi}{\mathrm{2}}\right) \\ $$

Question Number 146898 Answers: 2 Comments: 0

calculate ∫_0 ^∞ ((cosx)/((x^2 +1)(x^2 +2)(x^2 +3)))dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{\mathrm{cosx}}{\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{1}\right)\left(\mathrm{x}^{\mathrm{2}} +\mathrm{2}\right)\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{3}\right)}\mathrm{dx} \\ $$

Question Number 146705 Answers: 1 Comments: 0

find lim_(x→0) ((sin(tan(2x)−x)+1−cos(πx^2 ))/x^2 )

$$\mathrm{find}\:\mathrm{lim}_{\mathrm{x}\rightarrow\mathrm{0}} \:\frac{\mathrm{sin}\left(\mathrm{tan}\left(\mathrm{2x}\right)−\mathrm{x}\right)+\mathrm{1}−\mathrm{cos}\left(\pi\mathrm{x}^{\mathrm{2}} \right)}{\mathrm{x}^{\mathrm{2}} } \\ $$

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