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Question Number 60938 Answers: 0 Comments: 1

∫((csc^(2019) (x))/(sec^5 (x))) tan^2 (x) dx

$$\int\frac{{csc}^{\mathrm{2019}} \left({x}\right)}{{sec}^{\mathrm{5}} \left({x}\right)}\:{tan}^{\mathrm{2}} \left({x}\right)\:{dx} \\ $$

Question Number 60921 Answers: 1 Comments: 0

x^2 (d^2 y/dx^2 ) + x(dy/dx) + y=0 please solve this Euler equation

$${x}^{\mathrm{2}} \frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\:+\:{x}\frac{{dy}}{{dx}}\:+\:{y}=\mathrm{0} \\ $$$${please}\:{solve}\:{this}\:{Euler}\:{equation} \\ $$

Question Number 60915 Answers: 0 Comments: 0

Let Fibonacci sequence (F_n ) _(n≥0) where F_0 = 0, F_1 = 1, and F_(n+2) = F_(n+1) + F_n , ∀ n ≥ 0 . Find the least of natural numbers n so that F_n and F_(n+1) − 1 can be divided by F_(2019) .

$${Let}\:\:{Fibonacci}\:\:{sequence}\:\:\left({F}_{{n}} \right)\:_{{n}\geqslant\mathrm{0}} \\ $$$${where}\:\:{F}_{\mathrm{0}} \:=\:\mathrm{0},\:{F}_{\mathrm{1}} \:=\:\mathrm{1},\:\:{and}\:\:{F}_{{n}+\mathrm{2}} \:\:=\:\:{F}_{{n}+\mathrm{1}} \:+\:{F}_{{n}} \:\:\:\:,\:\:\forall\:{n}\:\:\geqslant\:\:\mathrm{0}\:. \\ $$$${Find}\:\:{the}\:\:{least}\:\:{of}\:\:{natural}\:\:{numbers}\:\:{n}\:\:{so}\:\:{that} \\ $$$${F}_{{n}} \:\:\:{and}\:\:\:{F}_{{n}+\mathrm{1}} \:−\:\mathrm{1}\:\:\:{can}\:\:{be}\:\:{divided}\:\:{by}\:\:\:{F}_{\mathrm{2019}} \:. \\ $$

Question Number 60906 Answers: 0 Comments: 1

Question Number 60905 Answers: 3 Comments: 0

if x+(1/x)=(√3).find x^(24) +x^(18) +x^6 +1

$${if}\:{x}+\frac{\mathrm{1}}{{x}}=\sqrt{\mathrm{3}}.{find} \\ $$$${x}^{\mathrm{24}} +{x}^{\mathrm{18}} +{x}^{\mathrm{6}} +\mathrm{1} \\ $$

Question Number 60901 Answers: 1 Comments: 0

find ∫ arctan((1/(1+x^2 )))dx

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

Question Number 60894 Answers: 0 Comments: 0

study the convergence of ∫_0 ^1 (((√(1+2x))−(√(1+x)))/(ln(1+x)))dx and determine its value.

$${study}\:{the}\:{convergence}\:{of}\: \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\sqrt{\mathrm{1}+\mathrm{2}{x}}−\sqrt{\mathrm{1}+{x}}}{{ln}\left(\mathrm{1}+{x}\right)}{dx}\:\:{and}\:{determine}\:{its} \\ $$$${value}. \\ $$

Question Number 60893 Answers: 1 Comments: 1

calculate ∫_0 ^(π/2) (dx/((√2)cos^2 x +(√3)sin^2 x))

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{dx}}{\sqrt{\mathrm{2}}{cos}^{\mathrm{2}} {x}\:+\sqrt{\mathrm{3}}{sin}^{\mathrm{2}} {x}} \\ $$

Question Number 60910 Answers: 1 Comments: 7

Question Number 60888 Answers: 1 Comments: 1

Question Number 60873 Answers: 2 Comments: 5

Question Number 60881 Answers: 0 Comments: 3

∫_(−π) ^π sin((1/(1−x^2 ))) dx

$$\underset{−\pi} {\overset{\pi} {\int}}{sin}\left(\frac{\mathrm{1}}{\mathrm{1}−{x}^{\mathrm{2}} }\right)\:{dx} \\ $$

Question Number 60862 Answers: 0 Comments: 0

Question Number 60854 Answers: 2 Comments: 4

(x^4 −3x^2 +2x+1)/(x−1)

$$\left({x}^{\mathrm{4}} −\mathrm{3}{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{1}\right)/\left({x}−\mathrm{1}\right) \\ $$$$ \\ $$$$ \\ $$

Question Number 60853 Answers: 2 Comments: 0

(√(5−12i))+(√(5+12i))=?

$$\sqrt{\mathrm{5}−\mathrm{12}{i}}+\sqrt{\mathrm{5}+\mathrm{12}{i}}=? \\ $$

Question Number 60849 Answers: 1 Comments: 5

For all θ in [0, π/2] show that cos(sinθ)≥sin(cosθ).

$${For}\:{all}\:\theta\:{in}\:\left[\mathrm{0},\:\pi/\mathrm{2}\right]\:{show}\:{that}\:{cos}\left({sin}\theta\right)\geqslant{sin}\left({cos}\theta\right). \\ $$

Question Number 61612 Answers: 0 Comments: 7

Question Number 60817 Answers: 4 Comments: 2

V=(4/3)𝛑R^3 prove

$$\boldsymbol{\mathrm{V}}=\frac{\mathrm{4}}{\mathrm{3}}\boldsymbol{\pi\mathrm{R}}^{\mathrm{3}} \:\:\:\boldsymbol{\mathrm{prove}} \\ $$

Question Number 60816 Answers: 1 Comments: 0

S=4𝛑R^2 prove

$$\boldsymbol{\mathrm{S}}=\mathrm{4}\boldsymbol{\pi\mathrm{R}}^{\mathrm{2}} \:\:\:\boldsymbol{\mathrm{prove}} \\ $$

Question Number 60814 Answers: 1 Comments: 0

find x given that 9^(sin^2 x) +9^(cos^2 x) =2

$${find}\:{x}\:{given}\:{that} \\ $$$$\mathrm{9}^{{sin}^{\mathrm{2}} {x}} +\mathrm{9}^{{cos}^{\mathrm{2}} {x}} =\mathrm{2}\: \\ $$$$ \\ $$

Question Number 60812 Answers: 0 Comments: 5

Question Number 60797 Answers: 0 Comments: 2

∫(e^w /w^(n+1) )dw, n∈N

$$\int\frac{{e}^{{w}} }{{w}^{{n}+\mathrm{1}} }{dw},\:{n}\in\mathbb{N} \\ $$

Question Number 60791 Answers: 1 Comments: 2

∫(e^n /x^(n+1) )dx, n∈N

$$\int\frac{{e}^{{n}} }{{x}^{{n}+\mathrm{1}} }{dx},\:\mathrm{n}\in\mathbb{N} \\ $$

Question Number 60788 Answers: 2 Comments: 1

Question Number 60783 Answers: 0 Comments: 2

∫_(−∞) ^∞ sin((1/(1+x^2 ))) dx

$$\underset{−\infty} {\overset{\infty} {\int}}{sin}\left(\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{2}} }\right)\:{dx} \\ $$

Question Number 60775 Answers: 0 Comments: 1

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