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Question Number 31061 Answers: 0 Comments: 0

find ∫_0 ^(π/2) (sinθ −cosθ)ln(sinθ+cosθ)dθ.

$${find}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\left({sin}\theta\:−{cos}\theta\right){ln}\left({sin}\theta+{cos}\theta\right){d}\theta. \\ $$

Question Number 31060 Answers: 0 Comments: 0

calculate by recurrence ∫_0 ^∞ ((lnx)/((1+x)^n ))dx with n≥2 .

$${calculate}\:{by}\:{recurrence}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{lnx}}{\left(\mathrm{1}+{x}\right)^{{n}} }{dx}\:{with}\:{n}\geqslant\mathrm{2}\:. \\ $$

Question Number 31059 Answers: 0 Comments: 0

find ∫_0 ^(π/2) cos(2θ)ln(tanθ)dθ.

$${find}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{cos}\left(\mathrm{2}\theta\right){ln}\left({tan}\theta\right){d}\theta. \\ $$

Question Number 31058 Answers: 0 Comments: 0

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

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

Question Number 31057 Answers: 0 Comments: 0

find ∫_0 ^1 (((√(1+x^2 )) −(√(1−x^2 )))/x^2 ) dx.

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\:−\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}{{x}^{\mathrm{2}} }\:{dx}.\: \\ $$

Question Number 31056 Answers: 0 Comments: 1

find ∫_1 ^(+∞) (dx/(x^2 −2xcosα +1)) with 0<α<π .

$${find}\:\:\int_{\mathrm{1}} ^{+\infty} \:\:\:\:\:\:\:\frac{{dx}}{{x}^{\mathrm{2}} \:−\mathrm{2}{xcos}\alpha\:+\mathrm{1}}\:\:{with}\:\mathrm{0}<\alpha<\pi\:. \\ $$

Question Number 31055 Answers: 0 Comments: 1

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

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

Question Number 31054 Answers: 0 Comments: 0

find ∫_0 ^1 (dx/(x^4 +1)) .

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dx}}{{x}^{\mathrm{4}} \:+\mathrm{1}}\:. \\ $$

Question Number 31053 Answers: 0 Comments: 1

let λ ∈R and a>0 find ∫_0 ^∞ e^(−ax) cos(λx)dx .

$${let}\:\lambda\:\in{R}\:{and}\:{a}>\mathrm{0}\:\:{find}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{ax}} {cos}\left(\lambda{x}\right){dx}\:. \\ $$

Question Number 31052 Answers: 0 Comments: 0

let give 0<a<b find ∫_a ^b ((lnx)/x)dx .

$${let}\:{give}\:\mathrm{0}<{a}<{b}\:\:{find}\:\int_{{a}} ^{{b}} \:\:\frac{{lnx}}{{x}}{dx}\:. \\ $$

Question Number 31051 Answers: 0 Comments: 0

study the convergence of ∫_0 ^∞ ((e^(−ax) −e^(−bx) )/(1− e^(−x) )) dx.

$${study}\:{the}\:{convergence}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{e}^{−{ax}} \:−{e}^{−{bx}} }{\mathrm{1}−\:{e}^{−{x}} }\:{dx}. \\ $$

Question Number 31049 Answers: 0 Comments: 0

study the convergence of ∫_0 ^∞ x^(−x) dx .

$${study}\:{the}\:{convergence}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:{x}^{−{x}} {dx}\:. \\ $$

Question Number 31048 Answers: 0 Comments: 0

study the convergence of ∫_1 ^(+∞) (((π/2) −arctanx)/x)dx

$${study}\:{the}\:{convergence}\:{of}\:\int_{\mathrm{1}} ^{+\infty} \:\:\frac{\frac{\pi}{\mathrm{2}}\:−{arctanx}}{{x}}{dx} \\ $$

Question Number 31047 Answers: 0 Comments: 0

let Δ={(x,y)∈N^2 /x+y=n , n∈N} find cardΔ 2) let A= {(x,y)∈N^2 / x+2y=n} find card A.

$${let}\:\Delta=\left\{\left({x},{y}\right)\in{N}^{\mathrm{2}} \:/{x}+{y}={n}\:,\:{n}\in{N}\right\}\:{find}\:{card}\Delta \\ $$$$\left.\mathrm{2}\right)\:{let}\:{A}=\:\left\{\left({x},{y}\right)\in{N}^{\mathrm{2}} /\:{x}+\mathrm{2}{y}={n}\right\}\:{find}\:{card}\:{A}. \\ $$

Question Number 31046 Answers: 0 Comments: 0

prove that C_n ^o C_n ^p +C_n ^1 C_(n−1) ^(p−1) +...C_n ^p C_(n−p) ^0 =2^p C_n ^p .

$${prove}\:{that}\:{C}_{{n}} ^{{o}} \:{C}_{{n}} ^{{p}} \:+{C}_{{n}} ^{\mathrm{1}} \:{C}_{{n}−\mathrm{1}} ^{{p}−\mathrm{1}} \:+...{C}_{{n}} ^{{p}} \:{C}_{{n}−{p}} ^{\mathrm{0}} =\mathrm{2}^{{p}} \:{C}_{{n}} ^{{p}} \:\:\:. \\ $$

Question Number 31045 Answers: 0 Comments: 0

simlify A_n =(C_n ^1 )^2 +2(C_n ^2 )^2 +n (C_n ^n )^2 .

$${simlify}\:{A}_{{n}} =\left({C}_{{n}} ^{\mathrm{1}} \right)^{\mathrm{2}} \:+\mathrm{2}\left({C}_{{n}} ^{\mathrm{2}} \right)^{\mathrm{2}} \:\:+{n}\:\left({C}_{{n}} ^{{n}} \right)^{\mathrm{2}} . \\ $$

Question Number 31044 Answers: 0 Comments: 0

if 1 +2^n +3^n +4^n =4q +r find r

$${if}\:\mathrm{1}\:+\mathrm{2}^{{n}} \:+\mathrm{3}^{{n}} \:+\mathrm{4}^{{n}} =\mathrm{4}{q}\:+{r}\:\:{find}\:{r}\: \\ $$

Question Number 31043 Answers: 0 Comments: 3

find n in ordre to have 7 divide n^3 +n−2

$${find}\:{n}\:{in}\:{ordre}\:{to}\:{have}\:\mathrm{7}\:{divide}\:{n}^{\mathrm{3}} \:+{n}−\mathrm{2} \\ $$

Question Number 31042 Answers: 0 Comments: 0

solve in N^2 9y^2 −(x+1)^2 =32 .

$${solve}\:{in}\:{N}^{\mathrm{2}} \:\:\mathrm{9}{y}^{\mathrm{2}} \:−\left({x}+\mathrm{1}\right)^{\mathrm{2}} =\mathrm{32}\:. \\ $$

Question Number 31041 Answers: 0 Comments: 0

solve in Z^2 11x −5y =14

$${solve}\:{in}\:{Z}^{\mathrm{2}} \:\mathrm{11}{x}\:−\mathrm{5}{y}\:=\mathrm{14} \\ $$

Question Number 31040 Answers: 0 Comments: 0

solve in Z^2 x^2 −y^2 =1969

$${solve}\:{in}\:{Z}^{\mathrm{2}} \:{x}^{\mathrm{2}} \:−{y}^{\mathrm{2}} =\mathrm{1969} \\ $$

Question Number 31039 Answers: 0 Comments: 0

find the sum s_0 = C_n ^o +C_n ^4 +C_n ^8 +... s_1 = C_n ^1 +C_n ^5 +C_n ^9 +.... s_3 = C_n ^2 +C_n ^6 + C_n ^(10) +....

$${find}\:{the}\:{sum} \\ $$$${s}_{\mathrm{0}} =\:{C}_{{n}} ^{{o}} \:+{C}_{{n}} ^{\mathrm{4}} \:+{C}_{{n}} ^{\mathrm{8}} \:+... \\ $$$${s}_{\mathrm{1}} =\:{C}_{{n}} ^{\mathrm{1}} \:+{C}_{{n}} ^{\mathrm{5}} \:+{C}_{{n}} ^{\mathrm{9}} \:+.... \\ $$$${s}_{\mathrm{3}} =\:{C}_{{n}} ^{\mathrm{2}} \:\:+{C}_{{n}} ^{\mathrm{6}} \:+\:{C}_{{n}} ^{\mathrm{10}} \:+.... \\ $$

Question Number 31038 Answers: 0 Comments: 0

let give α ∈]−π ,π[ 1)prove that sin^2 α −2(1+cosα) =−4cos^4 ((α/2)) 2)solve inside C z^2 −2z sinα +2(1+cosα)=0 find the module and arg of the roots.

$$\left.{let}\:{give}\:\alpha\:\in\right]−\pi\:,\pi\left[\right. \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:\:{sin}^{\mathrm{2}} \alpha\:−\mathrm{2}\left(\mathrm{1}+{cos}\alpha\right)\:=−\mathrm{4}{cos}^{\mathrm{4}} \left(\frac{\alpha}{\mathrm{2}}\right) \\ $$$$\left.\mathrm{2}\right){solve}\:{inside}\:{C}\:\:{z}^{\mathrm{2}} \:−\mathrm{2}{z}\:{sin}\alpha\:+\mathrm{2}\left(\mathrm{1}+{cos}\alpha\right)=\mathrm{0}\:{find} \\ $$$${the}\:{module}\:{and}\:{arg}\:{of}\:{the}\:{roots}. \\ $$

Question Number 31037 Answers: 0 Comments: 0

let θ∈[0,π] and Z=1+cosθ +isinθ Z^′ =cosθ +(1+sinθ)i find ∣ZZ^′ ∣ arg(ZZ^′ ) ∣(Z/Z^′ )∣ , arg((Z/Z^, )) , ∣Z−Z^, ∣ and arg(Z−Z^, ).

$${let}\:\:\theta\in\left[\mathrm{0},\pi\right]\:\:{and}\:{Z}=\mathrm{1}+{cos}\theta\:+{isin}\theta \\ $$$${Z}^{'} ={cos}\theta\:+\left(\mathrm{1}+{sin}\theta\right){i}\:\:{find}\:\mid{ZZ}^{'} \mid\:\:{arg}\left({ZZ}^{'} \right) \\ $$$$\mid\frac{{Z}}{{Z}^{'} }\mid\:,\:{arg}\left(\frac{{Z}}{{Z}^{,} }\right)\:\:,\:\:\mid{Z}−{Z}^{,} \mid\:{and}\:{arg}\left({Z}−{Z}^{,} \right). \\ $$

Question Number 31036 Answers: 0 Comments: 0

solve inside C z^6 = (z^− )^2 .

$${solve}\:{inside}\:{C}\:\:{z}^{\mathrm{6}} =\:\left({z}^{−} \right)^{\mathrm{2}} \:\:. \\ $$

Question Number 31035 Answers: 0 Comments: 0

1) solve x^5 =1 2) if x_i are roots of this equation prove that Σ_i x_i =0 3)prove that cos(((2π)/5)) +cos(((4π)/5)) =−(1/2) 4)calculate cos(((4π)/5)) interms cos(((2π)/5)) then find its values.

$$\left.\mathrm{1}\right)\:{solve}\:{x}^{\mathrm{5}} =\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{if}\:{x}_{{i}} \:{are}\:{roots}\:{of}\:{this}\:{equation}\:{prove}\:{that}\:\sum_{{i}} \:{x}_{{i}} =\mathrm{0} \\ $$$$\left.\mathrm{3}\right){prove}\:{that}\:{cos}\left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right)\:+{cos}\left(\frac{\mathrm{4}\pi}{\mathrm{5}}\right)\:=−\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\left.\mathrm{4}\right){calculate}\:{cos}\left(\frac{\mathrm{4}\pi}{\mathrm{5}}\right)\:{interms}\:{cos}\left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right)\:{then}\:{find}\:{its} \\ $$$${values}. \\ $$

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