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Question Number 52730 Answers: 0 Comments: 0

The area of a re

$${The}\:{area}\:{of}\:\:{a}\:{re} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 52731 Answers: 1 Comments: 11

Question Number 52728 Answers: 0 Comments: 0

How can you proove what follows ? If (4n+1) is prime, then : 4n + 1 = a^2 + b^2 with a, b ∈ N Thank you

$${How}\:{can}\:{you}\:{proove}\:{what}\:{follows}\:? \\ $$$$ \\ $$$$\mathrm{If}\:\left(\mathrm{4}{n}+\mathrm{1}\right)\:\mathrm{is}\:\mathrm{prime},\:\mathrm{then}\:: \\ $$$$\:\:\:\mathrm{4}{n}\:+\:\mathrm{1}\:=\:{a}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} \:\:\:\:\mathrm{with}\:{a},\:{b}\:\in\:\mathbb{N} \\ $$$$ \\ $$$${Thank}\:{you} \\ $$

Question Number 52713 Answers: 2 Comments: 1

if ∣x∣=∣y∣ is (x/y)=−1?

$${if}\:\mid{x}\mid=\mid{y}\mid\:{is}\:\frac{{x}}{{y}}=−\mathrm{1}? \\ $$

Question Number 52708 Answers: 0 Comments: 0

Question Number 52703 Answers: 0 Comments: 1

let f(t) =∫_0 ^∞ ((cos^2 (tx))/((x^2 +3)^2 )) dx with t ≥0 1) give a explicit form of f(t) 2) find the value of ∫_0 ^∞ ((xsin(2tx))/((x^2 +3)^2 )) dx 3) give the values of integrals ∫_0 ^∞ (dx/((x^2 +3)^2 )) and ∫_0 ^∞ ((cos^2 (πx))/((x^2 +3)^2 ))dx 4) give the values of integrals ∫_0 ^∞ ((xsin(πx))/((x^2 +3)^2 )) and ∫_0 ^∞ ((xsin(((πx)/2)))/((x^2 +3)^2 )) dx .

$${let}\:{f}\left({t}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}^{\mathrm{2}} \left({tx}\right)}{\left({x}^{\mathrm{2}} +\mathrm{3}\right)^{\mathrm{2}} }\:{dx}\:\:{with}\:{t}\:\geqslant\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{give}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({t}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{xsin}\left(\mathrm{2}{tx}\right)}{\left({x}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{2}} }\:{dx} \\ $$$$\left.\mathrm{3}\right)\:{give}\:{the}\:{values}\:{of}\:{integrals}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{2}} }\:{and}\:\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}^{\mathrm{2}} \left(\pi{x}\right)}{\left({x}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{4}\right)\:{give}\:{the}\:{values}\:{of}\:{integrals}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{xsin}\left(\pi{x}\right)}{\left({x}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{2}} }\:{and}\:\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{xsin}\left(\frac{\pi{x}}{\mathrm{2}}\right)}{\left({x}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{2}} }\:{dx}\:. \\ $$

Question Number 52683 Answers: 0 Comments: 3

let f(λ) =∫_(−∞) ^(+∞) ((sin(λx))/((x^2 +2λx +1)^2 ))dx with ∣λ∣<1 1) find the value of f(λ) 2) calculate ∫_(−∞) ^(+∞) ((sin((x/(2 ))))/((x^2 +x+1)^2 ))dx 3) find A(θ) =∫_(−∞) ^(+∞) ((sin((cosθ)x))/((x^2 +2cosθ x +1)^2 )) that we suppose 0<θ<(π/2)

$${let}\:{f}\left(\lambda\right)\:=\int_{−\infty} ^{+\infty} \:\:\frac{{sin}\left(\lambda{x}\right)}{\left({x}^{\mathrm{2}} \:+\mathrm{2}\lambda{x}\:+\mathrm{1}\right)^{\mathrm{2}} }{dx}\:\:\:\:{with}\:\mid\lambda\mid<\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{value}\:{of}\:{f}\left(\lambda\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{sin}\left(\frac{{x}}{\mathrm{2}\:}\right)}{\left({x}^{\mathrm{2}} \:\:+{x}+\mathrm{1}\right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{3}\right)\:{find}\:\:{A}\left(\theta\right)\:=\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{sin}\left(\left({cos}\theta\right){x}\right)}{\left({x}^{\mathrm{2}} \:+\mathrm{2}{cos}\theta\:{x}\:+\mathrm{1}\right)^{\mathrm{2}} }\:\:{that}\:{we}\:{suppose}\:\mathrm{0}<\theta<\frac{\pi}{\mathrm{2}} \\ $$

Question Number 52682 Answers: 1 Comments: 1

find nature of the serie Σ_(n=1) ^∞ (((√(n+1))−(√n))/(nln(n+1)))

$${find}\:{nature}\:{of}\:{the}\:{serie}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\sqrt{{n}+\mathrm{1}}−\sqrt{{n}}}{{nln}\left({n}+\mathrm{1}\right)} \\ $$

Question Number 52680 Answers: 0 Comments: 1

let f_n (x)=((sin(nx))/n^3 ) and f(x)=Σ_(n=1) ^∞ f_n (x) calculate ∫_0 ^π f(x)dx .

$${let}\:{f}_{{n}} \left({x}\right)=\frac{{sin}\left({nx}\right)}{{n}^{\mathrm{3}} }\:\:\:{and}\:{f}\left({x}\right)=\sum_{{n}=\mathrm{1}} ^{\infty} \:{f}_{{n}} \left({x}\right) \\ $$$${calculate}\:\int_{\mathrm{0}} ^{\pi} \:{f}\left({x}\right){dx}\:. \\ $$

Question Number 52679 Answers: 0 Comments: 1

let f_n (x)=(((−1)^n )/(n+x)) with x>0 1) study the simple convergence of Σ f_n (x) 2) calculate f^′ (x)

$${let}\:{f}_{{n}} \left({x}\right)=\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}+{x}}\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{study}\:{the}\:{simple}\:{convergence}\:{of}\:\Sigma\:{f}_{{n}} \left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{f}^{'} \left({x}\right) \\ $$

Question Number 52678 Answers: 1 Comments: 1

let u_n =ln{cos(2^(−n) )} calculate Σ_(n=0) ^∞ u_n

$${let}\:{u}_{{n}} ={ln}\left\{{cos}\left(\mathrm{2}^{−{n}} \right)\right\}\:\:{calculate}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:{u}_{{n}} \\ $$

Question Number 52677 Answers: 1 Comments: 1

find nature of Σ_(n=2) ^∞ (−1)^n (√n)ln(((n+1)/(n−1))).

$${find}\:{nature}\:{of}\:\sum_{{n}=\mathrm{2}} ^{\infty} \left(−\mathrm{1}\right)^{{n}} \sqrt{{n}}{ln}\left(\frac{{n}+\mathrm{1}}{{n}−\mathrm{1}}\right). \\ $$

Question Number 52675 Answers: 0 Comments: 1

let u_n =(−1)^n ∫_0 ^(π/2) sin^n xdx calculate Σ u_n

$${let}\:{u}_{{n}} =\left(−\mathrm{1}\right)^{{n}} \int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{sin}^{{n}} {xdx}\:\:{calculate}\:\Sigma\:{u}_{{n}} \\ $$

Question Number 52673 Answers: 1 Comments: 1

let f(x)=(x^n −1)e^x determine f^((n)) (x) with n integr natural

$${let}\:{f}\left({x}\right)=\left({x}^{{n}} −\mathrm{1}\right){e}^{{x}} \:\:\:{determine}\:{f}^{\left({n}\right)} \left({x}\right)\:\:\:{with}\:{n}\:{integr}\:{natural} \\ $$

Question Number 52671 Answers: 0 Comments: 1

study the sequence u_0 =1 and u_(n+1) =(1/(1+u_n ^2 ))

$${study}\:{the}\:{sequence}\:{u}_{\mathrm{0}} =\mathrm{1}\:{and}\:{u}_{{n}+\mathrm{1}} \:\:=\frac{\mathrm{1}}{\mathrm{1}+{u}_{{n}} ^{\mathrm{2}} } \\ $$

Question Number 52670 Answers: 1 Comments: 1

study the convergence of Σ_(n=0) ^∞ sin(π(√(4n^2 +1)))

$${study}\:{the}\:{convergence}\:{of}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:{sin}\left(\pi\sqrt{\mathrm{4}{n}^{\mathrm{2}} +\mathrm{1}}\right) \\ $$

Question Number 52669 Answers: 0 Comments: 0

let S_(n ) (p)=Σ_(k=0) ^n k^p prove that S_n (p)=(1/(p+1)){ (n+1)^(p+1) −Σ_(k=0) ^(n−1) C_(p+1) ^k S_n (k)}

$${let}\:{S}_{{n}\:} \:\left({p}\right)=\sum_{{k}=\mathrm{0}} ^{{n}} \:{k}^{{p}} \\ $$$${prove}\:{that}\:{S}_{{n}} \left({p}\right)=\frac{\mathrm{1}}{{p}+\mathrm{1}}\left\{\:\left({n}+\mathrm{1}\right)^{{p}+\mathrm{1}} \:−\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:{C}_{{p}+\mathrm{1}} ^{{k}} \:{S}_{{n}} \left({k}\right)\right\} \\ $$

Question Number 52667 Answers: 1 Comments: 0

∫((x^4 +1)/(x^2 (√(x^4 −1)))) dx

$$\int\frac{{x}^{\mathrm{4}} +\mathrm{1}}{{x}^{\mathrm{2}} \sqrt{{x}^{\mathrm{4}} −\mathrm{1}}}\:{dx} \\ $$

Question Number 52663 Answers: 2 Comments: 1

Question Number 52655 Answers: 0 Comments: 0

Question Number 52653 Answers: 0 Comments: 0

2 X + 5^(2 ) = (√9) − X 2 X + X = 3 − 25 3 X = − 22 X = ((22)/3) 6 Z − 63 = 1 − ∣ − 20 ∣ 6 Z = 1 − 20 + 63 6 Z = 44 Z = ((44)/6) = ((22)/3) ∵ s(x)=s(z),∴X=Z

$$\mathrm{2}\:{X}\:\:+\:\:\mathrm{5}\:^{\mathrm{2}\:} \:\:=\:\:\:\:\sqrt{\mathrm{9}}\:\:\:\:−\:\:\:{X} \\ $$$$\mathrm{2}\:\:{X}\:\:\:+\:\:\:{X}\:\:\:=\:\:\:\mathrm{3}\:\:\:−\:\:\:\mathrm{25} \\ $$$$\mathrm{3}\:\:{X}\:\:\:=\:\:\:−\:\mathrm{22} \\ $$$${X}\:\:=\:\:\frac{\mathrm{22}}{\mathrm{3}} \\ $$$$ \\ $$$$\mathrm{6}\:{Z}\:\:−\:\mathrm{63}\:\:=\:\:\mathrm{1}\:\:−\:\:\mid\:\:\:−\:\:\mathrm{20}\:\:\mid \\ $$$$\mathrm{6}\:{Z}\:\:=\:\:\mathrm{1}\:\:−\:\:\mathrm{20}\:\:+\:\:\mathrm{63} \\ $$$$\mathrm{6}\:\:{Z}\:\:=\:\:\mathrm{44} \\ $$$${Z}\:\:=\:\:\frac{\mathrm{44}}{\mathrm{6}}\:\:=\:\:\frac{\mathrm{22}}{\mathrm{3}}\:\:\:\:\:\because\:{s}\left({x}\right)={s}\left({z}\right),\therefore{X}={Z} \\ $$

Question Number 52649 Answers: 0 Comments: 2

∫ ((4x^2 + 3)/((x^2 + x + 1)^2 )) dx

$$\int\:\frac{\mathrm{4x}^{\mathrm{2}} \:+\:\mathrm{3}}{\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{x}\:+\:\mathrm{1}\right)^{\mathrm{2}} }\:\mathrm{dx} \\ $$

Question Number 52644 Answers: 0 Comments: 0

Question Number 52630 Answers: 0 Comments: 3

Question Number 52629 Answers: 2 Comments: 0

show that ((sinα+sin3α+sin5α)/(cosα+cos3α+cos5α))= tan3α

$${show}\:{that}\:\frac{{sin}\alpha+{sin}\mathrm{3}\alpha+{sin}\mathrm{5}\alpha}{{cos}\alpha+{cos}\mathrm{3}\alpha+{cos}\mathrm{5}\alpha}=\:{tan}\mathrm{3}\alpha \\ $$

Question Number 52627 Answers: 2 Comments: 1

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