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Question Number 173503 Answers: 0 Comments: 0

Question Number 173500 Answers: 2 Comments: 0

Find without any software: Ω = ∫ (x + (5/x))(1 − (5/x^2 ))sin(ln(x + (5/x)))dx

$$\mathrm{Find}\:\mathrm{without}\:\mathrm{any}\:\mathrm{software}: \\ $$$$\Omega\:=\:\int\:\left(\mathrm{x}\:+\:\frac{\mathrm{5}}{\mathrm{x}}\right)\left(\mathrm{1}\:−\:\frac{\mathrm{5}}{\mathrm{x}^{\mathrm{2}} }\right)\mathrm{sin}\left(\mathrm{ln}\left(\mathrm{x}\:+\:\frac{\mathrm{5}}{\mathrm{x}}\right)\right)\mathrm{dx} \\ $$

Question Number 173499 Answers: 0 Comments: 2

Question Number 173498 Answers: 1 Comments: 1

lim_(x→0) ((8cot (x)+9 sin ((1/x)))/(12 csc (x)−4sin ((1/x)))) =?

$$\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{8cot}\:\left({x}\right)+\mathrm{9}\:\mathrm{sin}\:\left(\frac{\mathrm{1}}{{x}}\right)}{\mathrm{12}\:\mathrm{csc}\:\left({x}\right)−\mathrm{4sin}\:\left(\frac{\mathrm{1}}{{x}}\right)}\:=? \\ $$

Question Number 173497 Answers: 0 Comments: 0

Bonus du Mardi 12/07/2022 I= ∫_0 ^(Π/2) ((sin^2 (x))/(1+cos^2 (x)))dx = ? J = ∫_0 ^(Π/2) (dx/(1+cos^2 (x))) I = ∫_0 ^(Π/2) ((2−(1+cos^2 (x))/(1+cos^2 (x)))dx = 2∫_0 ^(Π/2) (dx/(1+cos^2 (x)))−∫_0 ^(Π/2) dx . = 2J−(Π/2) determinant (((J=∫_0 ^(Π/2) (dx/(1+cos^2 (x))))),((I=2J−(Π/2)))) e

$$\:\:\: \\ $$$$\:\:{Bonus}\:{du}\:{Mardi}\:\mathrm{12}/\mathrm{07}/\mathrm{2022} \\ $$$$\:\:\:\:{I}=\:\:\int_{\mathrm{0}} ^{\frac{\Pi}{\mathrm{2}}} \:\frac{{sin}^{\mathrm{2}} \left({x}\right)}{\mathrm{1}+{cos}^{\mathrm{2}} \left({x}\right)}{dx}\:=\:? \\ $$$$\:\:\:\:\:\:\:\:\:\:{J}\:\:=\:\int_{\mathrm{0}} ^{\frac{\Pi}{\mathrm{2}}} \:\frac{{dx}}{\mathrm{1}+{cos}^{\mathrm{2}} \left({x}\right)} \\ $$$$\:{I}\:=\:\int_{\mathrm{0}} ^{\frac{\Pi}{\mathrm{2}}} \frac{\mathrm{2}−\left(\mathrm{1}+{cos}^{\mathrm{2}} \left({x}\right)\right.}{\mathrm{1}+{cos}^{\mathrm{2}} \left({x}\right)}{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:=\:\mathrm{2}\int_{\mathrm{0}} ^{\frac{\Pi}{\mathrm{2}}} \frac{{dx}}{\mathrm{1}+{cos}^{\mathrm{2}} \left({x}\right)}−\int_{\mathrm{0}} ^{\frac{\Pi}{\mathrm{2}}} \:{dx}\:.\:\: \\ $$$$=\:\mathrm{2}{J}−\frac{\Pi}{\mathrm{2}} \\ $$$$ \\ $$$$\begin{array}{|c|c|}{{J}=\int_{\mathrm{0}} ^{\frac{\Pi}{\mathrm{2}}} \frac{{dx}}{\mathrm{1}+{cos}^{\mathrm{2}} \left({x}\right)}}\\{{I}=\mathrm{2}{J}−\frac{\Pi}{\mathrm{2}}}\\\hline\end{array} \\ $$$$\:{e}\: \\ $$

Question Number 173493 Answers: 1 Comments: 0

find lim_(x→0) tan(tanhx)−tanh(tanx)

$$\boldsymbol{{find}}\:\boldsymbol{{lim}}_{\boldsymbol{{x}}\rightarrow\mathrm{0}} \:\boldsymbol{{tan}}\left(\boldsymbol{{tanhx}}\right)−\boldsymbol{{tanh}}\left(\boldsymbol{{tanx}}\right) \\ $$

Question Number 173489 Answers: 3 Comments: 0

Question Number 173486 Answers: 0 Comments: 1

𝛗 = ∫_0 ^( ∞) ((cos (ax)− sin(bx))/x^( 2) )dx =? 𝛗=∫_0 ^( ∞) ((1−2sin^( 2) (((ax)/2))−(1−2sin^( 2) (((bx)/2))))/(x^2 ))dx =2 {∫_0 ^( ∞) (((sin(((bx)/2)))/x))^2 dx=Θ_1 }−2{∫_0 ^( ∞) (((sin(((ax)/2)))/x))^2 dx=Θ_2 } Θ_( 1) =^(((bx)/2)=t) (b/2)∫_0 ^( ∞) ((sin^( 2) (t))/t^( 2) )dt= ((πb)/4) similarly : Θ_( 2) =((πa)/4) ∴ 𝛗= (π/2)(∣b∣−∣a∣) Dirichlet′s integrals: ∫_0 ^( ∞) ((sin(x))/x)dx=(π/2)=∫_0 ^( ∞) ((sin^2 (x))/x^( 2) )dx

$$ \\ $$$$\:\:\:\:\boldsymbol{\phi}\:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{{cos}\:\left({ax}\right)−\:{sin}\left({bx}\right)}{{x}^{\:\mathrm{2}} }{dx}\:\:=? \\ $$$$\:\:\: \\ $$$$\:\:\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\infty} \frac{\mathrm{1}−\mathrm{2}{sin}^{\:\mathrm{2}} \left(\frac{{ax}}{\mathrm{2}}\right)−\left(\mathrm{1}−\mathrm{2}{sin}^{\:\mathrm{2}} \left(\frac{{bx}}{\mathrm{2}}\right)\right)}{{x}\:^{\mathrm{2}} }{dx} \\ $$$$\:\:\:\:\:\:\:=\mathrm{2}\:\left\{\int_{\mathrm{0}} ^{\:\infty} \left(\frac{{sin}\left(\frac{{bx}}{\mathrm{2}}\right)}{{x}}\right)^{\mathrm{2}} {dx}=\Theta_{\mathrm{1}} \right\}−\mathrm{2}\left\{\int_{\mathrm{0}} ^{\:\infty} \left(\frac{{sin}\left(\frac{{ax}}{\mathrm{2}}\right)}{{x}}\right)^{\mathrm{2}} {dx}=\Theta_{\mathrm{2}} \right\} \\ $$$$\:\:\:\:\:\:\:\:\Theta_{\:\mathrm{1}} \overset{\frac{{bx}}{\mathrm{2}}={t}} {=}\:\frac{{b}}{\mathrm{2}}\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}^{\:\mathrm{2}} \left({t}\right)}{{t}^{\:\mathrm{2}} }{dt}=\:\frac{\pi{b}}{\mathrm{4}}\: \\ $$$$\:\:\:\:\:\:\:\:{similarly}\::\:\:\Theta_{\:\mathrm{2}} =\frac{\pi{a}}{\mathrm{4}}\:\:\:\:\:\therefore\:\:\:\boldsymbol{\phi}=\:\frac{\pi}{\mathrm{2}}\left(\mid{b}\mid−\mid{a}\mid\right)\: \\ $$$$\:\:\:\:\:{Dirichlet}'{s}\:{integrals}:\:\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}\left({x}\right)}{{x}}{dx}=\frac{\pi}{\mathrm{2}}=\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}^{\mathrm{2}} \left({x}\right)}{{x}^{\:\mathrm{2}} }{dx} \\ $$

Question Number 173485 Answers: 2 Comments: 0

Solve (x ∈ R ) x2^( (1/x)) +(1/x) 2^( x) = 4 −Source: L.Panaitopol

$$ \\ $$$$\:\:\:\:\:{Solve}\:\:\:\:\:\:\:\left({x}\:\in\:\mathbb{R}\:\right) \\ $$$$\:\:\:\:\:\:{x}\mathrm{2}^{\:\frac{\mathrm{1}}{{x}}} \:+\frac{\mathrm{1}}{{x}}\:\mathrm{2}^{\:{x}} =\:\mathrm{4} \\ $$$$\:\:\:\:\:\:\:\:−{Source}:\:{L}.{Panaitopol} \\ $$

Question Number 173484 Answers: 2 Comments: 0

1) lim_(x→0) ((x^(sinx) −(sinx)^x )/(x^(cosx) +1)) 2) lim_(x→∞) [x lnx − 2x ln(sin(1/( (√x)))) ] how can solve this proplem ?

$$\left.\mathrm{1}\right)\:\boldsymbol{{lim}}_{\boldsymbol{{x}}\rightarrow\mathrm{0}} \:\frac{\boldsymbol{{x}}^{\boldsymbol{{sinx}}} −\left(\boldsymbol{{sinx}}\right)^{\boldsymbol{{x}}} }{\boldsymbol{{x}}^{\boldsymbol{{cosx}}} +\mathrm{1}} \\ $$$$ \\ $$$$\left.\mathrm{2}\right)\:\boldsymbol{{lim}}_{\boldsymbol{{x}}\rightarrow\infty} \:\left[\boldsymbol{{x}}\:\boldsymbol{{lnx}}\:−\:\mathrm{2}\boldsymbol{{x}}\:\boldsymbol{{ln}}\left(\boldsymbol{{sin}}\frac{\mathrm{1}}{\:\sqrt{\boldsymbol{{x}}}}\right)\:\right] \\ $$$$ \\ $$$$\boldsymbol{{how}}\:\boldsymbol{{can}}\:\boldsymbol{{solve}}\:\boldsymbol{{this}}\:\boldsymbol{{proplem}}\:? \\ $$

Question Number 173445 Answers: 4 Comments: 0

Question Number 173431 Answers: 1 Comments: 2

Question Number 173430 Answers: 0 Comments: 3

Question Number 173426 Answers: 0 Comments: 0

In △ABC the following relationship holds: (a^3 +b^3 +c^3 )(h_a ^3 +h_b ^3 +h_c ^3 ) ≥ 5832 (√3) r^6

$$\mathrm{In}\:\:\bigtriangleup\mathrm{ABC}\:\:\mathrm{the}\:\mathrm{following}\:\mathrm{relationship} \\ $$$$\mathrm{holds}: \\ $$$$\left(\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} +\mathrm{c}^{\mathrm{3}} \right)\left(\mathrm{h}_{\boldsymbol{\mathrm{a}}} ^{\mathrm{3}} +\mathrm{h}_{\boldsymbol{\mathrm{b}}} ^{\mathrm{3}} +\mathrm{h}_{\boldsymbol{\mathrm{c}}} ^{\mathrm{3}} \right)\:\geqslant\:\mathrm{5832}\:\sqrt{\mathrm{3}}\:\mathrm{r}^{\mathrm{6}} \\ $$

Question Number 173424 Answers: 1 Comments: 0

prove that ∫_0 ^( ∞) (( sin(x))/(sinh(x))) dx = (π/2)tanh((π/2))

$$ \\ $$$$\:\:\:\:\mathrm{prove}\:\:\mathrm{that} \\ $$$$ \\ $$$$\:\:\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:\mathrm{sin}\left(\mathrm{x}\right)}{\mathrm{sinh}\left(\mathrm{x}\right)}\:\mathrm{dx}\:=\:\frac{\pi}{\mathrm{2}}\mathrm{tanh}\left(\frac{\pi}{\mathrm{2}}\right) \\ $$$$ \\ $$

Question Number 173421 Answers: 0 Comments: 0

In triangle ABC, AD is the bisector of ∠A meets BC at D, if 2∠C=∠B and AB=DC. Find ∠BAC

$$\mathrm{In}\:\mathrm{triangle}\:\:\mathrm{ABC},\:\mathrm{AD}\:\mathrm{is}\:\mathrm{the}\:\mathrm{bisector} \\ $$$$\mathrm{of}\:\:\angle\mathrm{A}\:\:\mathrm{meets}\:\:\mathrm{BC}\:\mathrm{at}\:\mathrm{D},\:\mathrm{if}\:\:\mathrm{2}\angle\mathrm{C}=\angle\mathrm{B} \\ $$$$\mathrm{and}\:\mathrm{AB}=\mathrm{DC}.\:\mathrm{Find}\:\angle\mathrm{BAC} \\ $$

Question Number 173420 Answers: 1 Comments: 0

Question Number 173418 Answers: 1 Comments: 0

Question Number 173417 Answers: 0 Comments: 1

Question Number 173447 Answers: 0 Comments: 4

Find tan (142.5°) without tables.

$${Find}\:\mathrm{tan}\:\left(\mathrm{142}.\mathrm{5}°\right)\:{without} \\ $$$${tables}. \\ $$

Question Number 173411 Answers: 1 Comments: 0

Q : ( 26 ! )^( 58) + k ≡^( 29) 0 and , k ∈ N , find: Min ( k )=?

$$ \\ $$$$\:\:\:\:\:\:\:\mathrm{Q}\:: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\left(\:\mathrm{26}\:!\:\right)^{\:\mathrm{58}} \:+\:\mathrm{k}\:\overset{\:\mathrm{29}} {\equiv}\:\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{and}\:,\:\:\mathrm{k}\:\in\:\mathbb{N}\:,\:\:\mathrm{find}:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Min}\:\left(\:\mathrm{k}\:\right)=? \\ $$$$ \\ $$

Question Number 173408 Answers: 1 Comments: 1

Question Number 173406 Answers: 0 Comments: 0

Find: Ω =lim_(n→∞) ((√((1 + n!)^(n!) ))/(n ∙ (n!)!))

$$\mathrm{Find}:\:\:\:\Omega\:=\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\:\frac{\sqrt{\left(\mathrm{1}\:+\:\mathrm{n}!\right)^{\boldsymbol{\mathrm{n}}!} }}{\mathrm{n}\:\centerdot\:\left(\mathrm{n}!\right)!} \\ $$

Question Number 173403 Answers: 0 Comments: 4

Question Number 173401 Answers: 0 Comments: 1

1.show that lim_(n→+∝) ((2^(2022) n^2 )/3^n )=0 ?

$$\mathrm{1}.\mathrm{show}\:\mathrm{that}\:\underset{\mathrm{n}\rightarrow+\propto} {\mathrm{lim}}\frac{\mathrm{2}^{\mathrm{2022}} \mathrm{n}^{\mathrm{2}} }{\mathrm{3}^{\mathrm{n}} }=\mathrm{0}\:? \\ $$

Question Number 173396 Answers: 1 Comments: 0

Consider the statements: x: Birds fly y: The sky is blue Which of the following statements can be represented aa x⇔y? (i) When the sky is blue, the birds fly. (ii) Either the bird is flying or the sky is blue. (iii) Birds fly if and only if the sky is blue. (iv) When birds fly, the sky is blue.

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