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The values of a for which y= ax^2 +ax+(1/(24)) and x = ay^2 +ay+(1/(24)) touch each other are 1) (2/3) 2) (3/2) 3) ((13+(√(601)))/(12)) 4) ((13−(√(601)))/(12)).

$$\mathrm{The}\:\mathrm{values}\:\mathrm{of}\:\mathrm{a}\:\mathrm{for}\:\mathrm{which}\:\mathrm{y}=\:\mathrm{a}{x}^{\mathrm{2}} +{ax}+\frac{\mathrm{1}}{\mathrm{24}} \\ $$$${and}\:{x}\:=\:{ay}^{\mathrm{2}} +{ay}+\frac{\mathrm{1}}{\mathrm{24}}\:{touch}\:{each}\:{other} \\ $$$${are} \\ $$$$\left.\mathrm{1}\left.\right)\:\frac{\mathrm{2}}{\mathrm{3}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}\right)\:\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$\left.\mathrm{3}\left.\right)\:\frac{\mathrm{13}+\sqrt{\mathrm{601}}}{\mathrm{12}}\:\:\:\:\:\:\:\mathrm{4}\right)\:\frac{\mathrm{13}−\sqrt{\mathrm{601}}}{\mathrm{12}}. \\ $$

Question Number 39383 Answers: 1 Comments: 1

calculate ∫_0 ^(π/3) ((sinxdx)/(cosx(2+ln(cosx))) .

$${calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{3}}} \:\:\:\:\:\:\frac{{sinxdx}}{{cosx}\left(\mathrm{2}+{ln}\left({cosx}\right)\right.}\:. \\ $$

Question Number 39382 Answers: 1 Comments: 0

Question Number 39381 Answers: 1 Comments: 0

Question Number 39380 Answers: 0 Comments: 0

find the value of Σ_(n=0) ^∞ (((−1)^n )/(4n+1))

$${find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\mathrm{4}{n}+\mathrm{1}} \\ $$

Question Number 39379 Answers: 1 Comments: 6

Question Number 39378 Answers: 0 Comments: 1

study the derivability of f(x)=Σ_(n=0) ^∞ (((−1)^n )/(nx +1))

$${study}\:{the}\:{derivability}\:{of} \\ $$$${f}\left({x}\right)=\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{nx}\:+\mathrm{1}} \\ $$

Question Number 39376 Answers: 0 Comments: 0

how to calculate the product (Σ_(n=0) ^∞ a_n x^n ).(Σ_(n=0) ^∞ b_n x^(2n) )?

$${how}\:{to}\:{calculate}\:{the}\:{product}\:\left(\sum_{{n}=\mathrm{0}} ^{\infty} {a}_{{n}} {x}^{{n}} \right).\left(\sum_{{n}=\mathrm{0}} ^{\infty} \:{b}_{{n}} \:{x}^{\mathrm{2}{n}} \right)? \\ $$

Question Number 39375 Answers: 0 Comments: 1

Question Number 39374 Answers: 0 Comments: 2

calculate I = ∫_0 ^∞ ((arctan(x^2 ))/(1+x^2 ))dx

$${calculate}\:{I}\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{arctan}\left({x}^{\mathrm{2}} \right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 39373 Answers: 0 Comments: 1

find the values of integrals A = ∫_(−∞) ^(+∞) cos(x^2 +x+1)dx and B = ∫_(−∞) ^(+∞) sin(x^2 +x+1)dx

$${find}\:{the}\:{values}\:{of}\:{integrals} \\ $$$${A}\:=\:\int_{−\infty} ^{+\infty} \:{cos}\left({x}^{\mathrm{2}} \:+{x}+\mathrm{1}\right){dx}\:\:\:{and}\:{B}\:=\:\int_{−\infty} ^{+\infty} \:{sin}\left({x}^{\mathrm{2}} \:+{x}+\mathrm{1}\right){dx} \\ $$

Question Number 39371 Answers: 2 Comments: 1

Question Number 39370 Answers: 0 Comments: 1

let I (λ) = ∫_(−∞) ^(+∞) ((cos(λx))/((1+ix)^2 ))dx 1) extract Re(I(λ)) and Im(I(λ)) 2) calculate I(λ) 3) conclude the values of Re(I(λ)) and Im(I(λ)).

$${let}\:{I}\:\left(\lambda\right)\:=\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{cos}\left(\lambda{x}\right)}{\left(\mathrm{1}+{ix}\right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{1}\right)\:\:{extract}\:{Re}\left({I}\left(\lambda\right)\right)\:{and}\:{Im}\left({I}\left(\lambda\right)\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{I}\left(\lambda\right) \\ $$$$\left.\mathrm{3}\right)\:{conclude}\:\:{the}\:{values}\:{of}\:{Re}\left({I}\left(\lambda\right)\right)\:{and}\:{Im}\left({I}\left(\lambda\right)\right). \\ $$

Question Number 39369 Answers: 0 Comments: 1

1) calculate F(x)= ∫_1 ^(√x) ((arctan(t))/t^2 )dt with x≥1 2) calculate A_n = ∫_1 ^(√n) ((arctan(t))/t^2 ) dt and find lim_(n→+∞) A_n

$$\left.\mathrm{1}\right)\:{calculate}\:{F}\left({x}\right)=\:\int_{\mathrm{1}} ^{\sqrt{{x}}} \:\:\:\frac{{arctan}\left({t}\right)}{{t}^{\mathrm{2}} }{dt}\:\:\:{with}\:{x}\geqslant\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\:{A}_{{n}} =\:\int_{\mathrm{1}} ^{\sqrt{{n}}} \:\:\frac{{arctan}\left({t}\right)}{{t}^{\mathrm{2}} }\:{dt}\:\:{and}\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} \\ $$

Question Number 39368 Answers: 0 Comments: 1

let F(t)= ∫_0 ^(+∞) ((sinx)/(x(1+x^2 ))) e^(−tx(1+x^2 )) dx witht≥0 1) caculate (dF/dt)(t) 2) find a simple form of F(t) 3) find the value of ∫_0 ^∞ ((sinx)/(x(1+x^2 )dx )).

$${let}\:{F}\left({t}\right)=\:\int_{\mathrm{0}} ^{+\infty} \:\:\:\frac{{sinx}}{{x}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}\:{e}^{−{tx}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)} {dx}\:\:{witht}\geqslant\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{caculate}\:\:\frac{{dF}}{{dt}}\left({t}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{a}\:{simple}\:{form}\:{of}\:{F}\left({t}\right) \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{sinx}}{{x}\left(\mathrm{1}+{x}^{\mathrm{2}} \right){dx}\:}. \\ $$

Question Number 39365 Answers: 0 Comments: 1

Question Number 39357 Answers: 0 Comments: 0

∫ (1/(xln(x+1))) dx

$$\int\:\frac{\mathrm{1}}{{xln}\left({x}+\mathrm{1}\right)}\:{dx} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 39349 Answers: 1 Comments: 1

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