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let ϕ(a) =∫_1 ^(√3) arctan((a/x))dx 1) calculate ϕ(a) interms of a 2) calculate ϕ^′ (a) at form of integral. 3) determine ϕ^((n)) (a) at form of integral. 4) find the value of ∫_1 ^(√3) arctan((2/x))dx .

$${let}\:\varphi\left({a}\right)\:=\int_{\mathrm{1}} ^{\sqrt{\mathrm{3}}} \:\:{arctan}\left(\frac{{a}}{{x}}\right){dx} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\varphi\left({a}\right)\:{interms}\:{of}\:{a} \\ $$$$\left.\mathrm{2}\right)\:\:{calculate}\:\varphi^{'} \left({a}\right)\:{at}\:{form}\:{of}\:{integral}. \\ $$$$\left.\mathrm{3}\right)\:{determine}\:\varphi^{\left({n}\right)} \left({a}\right)\:\:{at}\:{form}\:{of}\:{integral}. \\ $$$$\left.\mathrm{4}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{1}} ^{\sqrt{\mathrm{3}}} \:{arctan}\left(\frac{\mathrm{2}}{{x}}\right){dx}\:. \\ $$

Question Number 55273 Answers: 0 Comments: 1

let f(x) =∫_x^2 ^(1+x) (dt/(1+t+t^2 )) 1) calculate f(x) interms of x 2) calculate lim_(x→0) f(x) and lim_(x→+∞) f(x)

$${let}\:\:{f}\left({x}\right)\:=\int_{{x}^{\mathrm{2}} } ^{\mathrm{1}+{x}} \:\:\frac{{dt}}{\mathrm{1}+{t}+{t}^{\mathrm{2}} } \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({x}\right)\:{interms}\:{of}\:{x} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} {f}\left({x}\right)\:{and}\:{lim}_{{x}\rightarrow+\infty} \:\:{f}\left({x}\right) \\ $$

Question Number 55271 Answers: 0 Comments: 0

1) let f(x) =∫_0 ^(2π) ((cost)/(3 +sin(xt)))dt find a explicit form of f(x) 2) calculate g(x) =∫_0 ^(2π) ((tcos(xt)cost)/((3 +sin(xt))^2 ))dt 3) calculate ∫_0 ^(2π) ((cost)/(3+sint)) and ∫_0 ^(2π) ((t cos^2 t)/((3+sint)^2 ))dt

$$\left.\mathrm{1}\right)\:{let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\frac{{cost}}{\mathrm{3}\:+{sin}\left({xt}\right)}{dt} \\ $$$${find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\frac{{tcos}\left({xt}\right){cost}}{\left(\mathrm{3}\:+{sin}\left({xt}\right)\right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{cost}}{\mathrm{3}+{sint}}\:\:{and}\:\:\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{t}\:{cos}^{\mathrm{2}} {t}}{\left(\mathrm{3}+{sint}\right)^{\mathrm{2}} }{dt} \\ $$

Question Number 55270 Answers: 0 Comments: 1

let f(x)=x^n arctan(x^2 ) with n integr natural 1) calculate f^((n)) (x) and f^((n)) (0) 2)developp f at integr serie .

$${let}\:{f}\left({x}\right)={x}^{{n}} {arctan}\left({x}^{\mathrm{2}} \right)\:\:{with}\:{n}\:{integr}\:{natural} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({n}\right)} \left({x}\right)\:{and}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right){developp}\:{f}\:{at}\:{integr}\:{serie}\:. \\ $$$$ \\ $$

Question Number 55269 Answers: 0 Comments: 4

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

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

Question Number 55268 Answers: 0 Comments: 1

calculate Σ_(n=0) ^∞ (((−1)^n )/(4n+3))

$${calculate}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\mathrm{4}{n}+\mathrm{3}} \\ $$

Question Number 55308 Answers: 1 Comments: 0

Question Number 55288 Answers: 2 Comments: 0

Question Number 55258 Answers: 0 Comments: 0

3x+5y=?_

$$\mathrm{3}{x}+\mathrm{5}{y}=?_{} \\ $$

Question Number 55267 Answers: 0 Comments: 0

1) calculate f(x)=∫_0 ^(π/4) ln(1+xtanθ)dθ 2) find the values of integrals ∫_0 ^(π/4) ln(1+tanθ) and ∫_0 ^(π/4) ln(1+2tanθ)dθ . 1) we have f^′ (x)=∫_0 ^(π/4) ((tanθ)/(1+xtanθ)) dθ =∫_0 ^(π/4) (((sinθ)/(cosθ))/(1+x((sinθ)/(cosθ))))dθ =∫_0 ^(π/4) ((sinθ)/(cosθ +xsinθ)) dθ =_(tan((θ/2))=t) ∫_0 ^((√2)−1) (((2t)/(1+t^2 ))/(((1−t^2 )/(1+t^2 )) +((2xt)/(1+t^2 )))) ((2dt)/(1+t^2 )) =∫_0 ^((√2)−1) ((4t)/((1+t^2 )(1−t^2 +2xt)))dt =−∫_0 ^((√2)−1) ((4t)/((t^2 +1)(t^2 −2xt −1)))dt let decompose F(t) = ((4t)/((t^2 +1)(t^2 −2xt −1))) roots of t^2 −2xt −1 Δ^′ =x^2 +1 ⇒t_1 =x+(√(x^2 +1)) and t_2 =x−(√(x^2 +1)) F(t)=(a/(t−t_1 )) +(b/(t−t_2 )) +((ct +d)/(t^2 +1)) a =lim_(t→t_1 ) (t−t_1 )F(t)=((4t_1 )/((t_1 ^2 +1)(t_1 −t_2 ))) =α b =lim_(t→t_2 ) (t−t_2 )F(t) =((4t_2 )/((t_2 ^2 +1)(t_2 −t_1 ))) =β ⇒F(t)=(α/(t−t_1 )) +(β/(t−t_2 )) +((ct +d)/(t^2 +1)) F(0) =0=−(α/t_1 ) −(β/t_2 ) +d ⇒d =(α/t_1 ) +(β/t_2 ) F(1)=(2/(−2x)) =−(1/x)=(α/(1−t_1 )) +(β/(1−t_2 )) +((c+d)/2) ⇒(1/x) =(α/(t_1 −1)) +(β/(t_2 −1)) −(c/2) −(d/2) ⇒(c/2) =(α/(t_1 −1)) +(β/(t_2 −1)) −(d/2) −(1/x) ⇒c =((2α)/(t_1 −1)) +((2β)/(t_2 −1)) −d−(2/x) ∫ F(t)dt =αln∣t−t_1 ∣ +βln∣t−t_2 ∣ +(c/2)ln(t^2 +1) +d arctan(t) ⇒ ∫_0 ^((√2)−1) F(t)dt =[αln∣t−t_1 ∣+βln∣t−t_2 ∣ +(c/2)ln(t^2 +1)]_0 ^((√2)−1) =αln∣(√2)−1−t_1 ∣ +βln∣(√2)−1−t_2 ∣ +(c/2)ln(4−2(√2)) =αln∣(√2)−1−x−(√(1+x^2 )))+βln∣(√2)−1−x+(√(1+x^2 ))) +((ln(4−2(√2)))/2)c =f^′ (x) ⇒ f(x)=∫ αln∣(√2)−1−x−(√(1+x^2 ))∣)dx+β∫ ln∣(√2)−1+(√(1+x^2 ))∣dx +((cx)/2)ln(4−2(√2)) +C ....be continued...

Question Number 55296 Answers: 1 Comments: 0

simplify the following expression if x<0 ∣4x−(√((3x−1)^2 ))∣

$${simplify}\:{the}\:{following}\:{expression}\:{if} \\ $$$${x}<\mathrm{0} \\ $$$$\mid\mathrm{4}{x}−\sqrt{\left(\mathrm{3}{x}−\mathrm{1}\right)^{\mathrm{2}} }\mid \\ $$

Question Number 55295 Answers: 0 Comments: 0