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let f(x)=∫_0 ^(2π) ((sint)/(x +sint))dt withx>1 1) calculate f(x) 2) calculate ∫_0 ^(2π) ((sint)/((x+sint)^2 ))dt 3)find the value of ∫_0 ^(2π) ((sint)/(2+sint))dt and ∫_0 ^(2π) ((sint)/((2+sint)^2 ))dt

$${let}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{sint}}{{x}\:+{sint}}{dt}\:\:{withx}>\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{sint}}{\left({x}+{sint}\right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{sint}}{\mathrm{2}+{sint}}{dt}\:{and}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{sint}}{\left(\mathrm{2}+{sint}\right)^{\mathrm{2}} }{dt} \\ $$

Question Number 46850 Answers: 1 Comments: 1

let a^2 >b^(2 ) +c^2 calculate ∫_0 ^(2π) (dθ/(a+bsinθ +c cosθ))

$${let}\:{a}^{\mathrm{2}} >{b}^{\mathrm{2}\:} +{c}^{\mathrm{2}} \:\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{d}\theta}{{a}+{bsin}\theta\:+{c}\:{cos}\theta} \\ $$

Question Number 46849 Answers: 0 Comments: 0

let A_p =Σ_(n=1) ^∞ n^p x^n with p integr . and x ∈]−1,1[ . 1) calculate A_1 ,A_2 and A_3 2) find a relation of recurrence betwen the A_n 3) calculate Σ_(n=1) ^∞ n^4 x^n and Σ_(n=1) ^∞ n^5 x^n .

$$\left.{let}\:{A}_{{p}} =\sum_{{n}=\mathrm{1}} ^{\infty} \:{n}^{{p}} {x}^{{n}} \:\:\:\:{with}\:{p}\:{integr}\:.\:{and}\:{x}\:\in\right]−\mathrm{1},\mathrm{1}\left[\:.\right. \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}_{\mathrm{1}} ,{A}_{\mathrm{2}} \:{and}\:{A}_{\mathrm{3}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{a}\:{relation}\:{of}\:{recurrence}\:\:{betwen}\:{the}\:{A}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:{n}^{\mathrm{4}} {x}^{{n}} \:{and}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:{n}^{\mathrm{5}} {x}^{{n}} \:. \\ $$

Question Number 46848 Answers: 0 Comments: 1

caculate ∫∫_D (x^2 −y^2 ) e^(−x^2 −y^2 ) dxdy with D ={(x,y)∈R^2 / x^2 +y^2 ≤4}

$${caculate}\:\:\int\int_{{D}} \:\:\left({x}^{\mathrm{2}} −{y}^{\mathrm{2}} \right)\:{e}^{−{x}^{\mathrm{2}} \:−{y}^{\mathrm{2}} } {dxdy}\:\:{with} \\ $$$${D}\:=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /\:\:\:{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:\leqslant\mathrm{4}\right\} \\ $$

Question Number 46847 Answers: 0 Comments: 1

calculate ∫∫_(0≤x≤1 and 1≤y≤2) e^(x/y) dxdy

$${calculate}\:\:\int\int_{\mathrm{0}\leqslant{x}\leqslant\mathrm{1}\:{and}\:\mathrm{1}\leqslant{y}\leqslant\mathrm{2}} \:\:{e}^{\frac{{x}}{{y}}} {dxdy} \\ $$

Question Number 46846 Answers: 0 Comments: 1

calculate ∫∫_D ((x+y)/(√(1−x^2 −y^2 )))dxdy with D={(x,y)∈R^2 /x≥0,y≥0,x^2 +y^2 <1}

$${calculate}\:\int\int_{{D}} \:\:\:\:\frac{{x}+{y}}{\sqrt{\mathrm{1}−{x}^{\mathrm{2}} −{y}^{\mathrm{2}} }}{dxdy}\:{with}\:{D}=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /{x}\geqslant\mathrm{0},{y}\geqslant\mathrm{0},{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} <\mathrm{1}\right\} \\ $$

Question Number 46845 Answers: 0 Comments: 0

calculate ∫_0 ^1 (e^(−x) /(1+x)) dx .