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Question Number 79106 Answers: 1 Comments: 2

calculate ∫_0 ^∞ e^(−(x^2 +(1/x^2 ))) dx

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

Question Number 79105 Answers: 0 Comments: 0

caculate ∫_0 ^∞ (dx/((x+1)(x+2)....(x+n))) with n integr ≥2

$${caculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{\left({x}+\mathrm{1}\right)\left({x}+\mathrm{2}\right)....\left({x}+{n}\right)}\:\:{with}\:{n}\:{integr}\:\geqslant\mathrm{2} \\ $$

Question Number 79104 Answers: 0 Comments: 0

decompose F(x)=(1/((x^2 −1)(x^2 −2^2 )....(x^2 −n^2 ))) inside R(x)

$${decompose}\:{F}\left({x}\right)=\frac{\mathrm{1}}{\left({x}^{\mathrm{2}} −\mathrm{1}\right)\left({x}^{\mathrm{2}} −\mathrm{2}^{\mathrm{2}} \right)....\left({x}^{\mathrm{2}} −{n}^{\mathrm{2}} \right)}\:{inside}\:{R}\left({x}\right) \\ $$

Question Number 79103 Answers: 0 Comments: 1

calculate lim_(x→1) ((nx^(n+1) −(n+1)x^n +1)/((x−1)^2 )) without hospital rule.

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{1}} \:\:\frac{{nx}^{{n}+\mathrm{1}} −\left({n}+\mathrm{1}\right){x}^{{n}} \:+\mathrm{1}}{\left({x}−\mathrm{1}\right)^{\mathrm{2}} }\:\:{without}\:{hospital}\:{rule}. \\ $$

Question Number 79102 Answers: 0 Comments: 1

calculate lim_(x→0) ((ln(1+e^(−x^2 ) )−ln(2))/x^2 )

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\frac{{ln}\left(\mathrm{1}+{e}^{−{x}^{\mathrm{2}} } \right)−{ln}\left(\mathrm{2}\right)}{{x}^{\mathrm{2}} } \\ $$

Question Number 79101 Answers: 0 Comments: 1

calculate lim_(x→0) ((sin(e^(−x^2 ) )+sinx−sin(1))/x^3 )

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\:\frac{{sin}\left({e}^{−{x}^{\mathrm{2}} } \right)+{sinx}−{sin}\left(\mathrm{1}\right)}{{x}^{\mathrm{3}} } \\ $$

Question Number 79100 Answers: 0 Comments: 1

calculate ∫_0 ^∞ (((−1)^x^2 )/((x^2 −x+1)^3 ))dx

$${calculate}\:\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{x}^{\mathrm{2}} } }{\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right)^{\mathrm{3}} }{dx} \\ $$

Question Number 79098 Answers: 0 Comments: 0

calculate Σ_(n=2) ^∞ (((−1)^n )/(n^4 −1))

$${calculate}\:\sum_{{n}=\mathrm{2}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{4}} −\mathrm{1}} \\ $$

Question Number 79097 Answers: 0 Comments: 0

find Σ_(n=2) ^∞ (1/(n^4 −1))

$${find}\:\sum_{{n}=\mathrm{2}} ^{\infty} \:\:\frac{\mathrm{1}}{{n}^{\mathrm{4}} −\mathrm{1}} \\ $$

Question Number 79096 Answers: 1 Comments: 1

calculate ∫_0 ^∞ ((ln(x))/((1+x)^3 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}\left({x}\right)}{\left(\mathrm{1}+{x}\right)^{\mathrm{3}} }{dx} \\ $$

Question Number 79095 Answers: 0 Comments: 1

find A_n =∫_0 ^∞ ((sin(x)sin(2x)....sin(nx))/x^n )dx with n≥2 integr

$${find}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sin}\left({x}\right){sin}\left(\mathrm{2}{x}\right)....{sin}\left({nx}\right)}{{x}^{{n}} }{dx}\:\:{with}\:{n}\geqslant\mathrm{2}\:{integr} \\ $$

Question Number 79094 Answers: 1 Comments: 0

find I_(a,b) =∫_0 ^∞ ((sin(ax)sin(bx))/x^2 )dx witha>0 and b>0

$${find}\:{I}_{{a},{b}} \:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sin}\left({ax}\right){sin}\left({bx}\right)}{{x}^{\mathrm{2}} }{dx}\:\:\:{witha}>\mathrm{0}\:{and}\:{b}>\mathrm{0} \\ $$

Question Number 79093 Answers: 0 Comments: 0

find f(λ) =∫_0 ^∞ e^(−λx^2 ) ch(x^2 +λ)dx with λ>0

$${find}\:\:{f}\left(\lambda\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−\lambda{x}^{\mathrm{2}} } {ch}\left({x}^{\mathrm{2}} \:+\lambda\right){dx}\:\:{with}\:\lambda>\mathrm{0} \\ $$

Question Number 79092 Answers: 0 Comments: 0

find ∫_(−∞) ^(+∞) ((e^(−x^2 ) arctan(x^2 +1))/(x^2 +1))dx

$${find}\:\int_{−\infty} ^{+\infty} \:\:\frac{{e}^{−{x}^{\mathrm{2}} } {arctan}\left({x}^{\mathrm{2}} +\mathrm{1}\right)}{{x}^{\mathrm{2}} \:+\mathrm{1}}{dx} \\ $$

Question Number 79091 Answers: 0 Comments: 0

calculate ∫_0 ^∞ ((e^(−x^2 ) arctan(x))/x)dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{x}^{\mathrm{2}} } \:{arctan}\left({x}\right)}{{x}}{dx} \\ $$

Question Number 79089 Answers: 0 Comments: 2

Question Number 79086 Answers: 0 Comments: 2

if:∫cos(f(x))dx=g(x) ∫sin(f(x))dx=? (use g(x))

$$\mathrm{if}:\int\mathrm{cos}\left(\mathrm{f}\left(\mathrm{x}\right)\right)\mathrm{dx}=\mathrm{g}\left(\mathrm{x}\right) \\ $$$$\int\mathrm{sin}\left(\mathrm{f}\left(\mathrm{x}\right)\right)\mathrm{dx}=?\:\left(\mathrm{use}\:\mathrm{g}\left(\mathrm{x}\right)\right) \\ $$

Question Number 79085 Answers: 1 Comments: 1

α and β (or more) are root of: (x^2 +x)^2 −2(x^2 +x)−k=0 and αβ<0 ⇒k∈?

$$\alpha\:\mathrm{and}\:\beta\:\left(\mathrm{or}\:\mathrm{more}\right)\:\:\mathrm{are}\:\mathrm{root}\:\mathrm{of}: \\ $$$$\left(\mathrm{x}^{\mathrm{2}} +\mathrm{x}\right)^{\mathrm{2}} −\mathrm{2}\left(\mathrm{x}^{\mathrm{2}} +\mathrm{x}\right)−\mathrm{k}=\mathrm{0} \\ $$$$\mathrm{and}\:\:\alpha\beta<\mathrm{0}\:\:\:\:\:\:\:\:\Rightarrow\mathrm{k}\in? \\ $$

Question Number 79081 Answers: 1 Comments: 0

I dont know how to show the functions of this type if it is continuous or not in topological space. I know the method is to use one of two properties: the inverse image of eah open is open, and the inverse image of each closed is closed but i dont know how to use them. i will be thaked if someone help ed me with the steps if f:(R,U) −> (R,U) defined by f(x) = { (((1/3) x +1 , x>3)),(((1/2) (x+5) , x≦3)) :} Is f continuous?

$${I}\:{dont}\:{know}\:{how}\:{to}\:{show}\:{the}\:{functions}\:{of}\: \\ $$$${this}\:{type}\:{if}\:{it}\:{is}\:{continuous}\:{or}\:{not}\:{in}\:{topological}\:{space}. \\ $$$${I}\:{know}\:{the}\:{method}\:{is}\:{to}\:{use}\:{one}\:{of}\:{two}\:{properties}: \\ $$$${the}\:{inverse}\:{image}\:{of}\:{eah}\:{open}\:{is}\:{open}, \\ $$$${and}\:{the}\:{inverse}\:{image}\:{of}\:{each}\:{closed}\:{is}\:{closed} \\ $$$${but}\:{i}\:{dont}\:{know}\:{how}\:{to}\:{use}\:{them}. \\ $$$${i}\:{will}\:{be}\:{thaked}\:{if}\:{someone}\:{help}\:{ed}\:{me}\:{with}\:{the}\:{steps} \\ $$$$ \\ $$$${if}\:{f}:\left(\mathbb{R},{U}\right)\:−>\:\left(\mathbb{R},{U}\right)\:{defined}\:{by} \\ $$$$ \\ $$$${f}\left({x}\right)\:=\:\begin{cases}{\frac{\mathrm{1}}{\mathrm{3}}\:{x}\:+\mathrm{1}\:\:\:,\:{x}>\mathrm{3}}\\{\frac{\mathrm{1}}{\mathrm{2}}\:\left({x}+\mathrm{5}\right)\:\:,\:{x}\leqq\mathrm{3}}\end{cases} \\ $$$${Is}\:{f}\:{continuous}? \\ $$

Question Number 79079 Answers: 1 Comments: 0

Question Number 79075 Answers: 1 Comments: 0

Hello solve in [0;2π] tan2x≥(√3)

$$\mathrm{Hello}\: \\ $$$$\mathrm{solve}\:\mathrm{in}\:\left[\mathrm{0};\mathrm{2}\pi\right]\:\mathrm{tan2}{x}\geqslant\sqrt{\mathrm{3}} \\ $$

Question Number 79065 Answers: 0 Comments: 5

]. lim_(x→0) (([sin(α+β)x+sin(α−β)x+sin2αx)/(cos2βx−cos2αx)).x

$$\left.\right].\:{li}\underset{{x}\rightarrow\mathrm{0}} {{m}}\:\frac{\left[{sin}\left(\alpha+\beta\right){x}+{sin}\left(\alpha−\beta\right){x}+{sin}\mathrm{2}\alpha{x}\right.}{{cos}\mathrm{2}\beta{x}−{cos}\mathrm{2}\alpha{x}}.{x} \\ $$

Question Number 79064 Answers: 1 Comments: 0

find lim_(x→ −3) ((x^3 +27)/(x^5 +243))

$${find}\:\:{li}\underset{{x}\rightarrow\:−\mathrm{3}} {{m}}\frac{{x}^{\mathrm{3}} +\mathrm{27}}{{x}^{\mathrm{5}} +\mathrm{243}} \\ $$

Question Number 79062 Answers: 1 Comments: 0

if a is a rational number with ∣a∣≤1, prove that cos (n cos^(−1) (a)) is also a rational number. (n∈N)

$${if}\:{a}\:{is}\:{a}\:{rational}\:{number}\:{with}\:\mid{a}\mid\leqslant\mathrm{1}, \\ $$$${prove}\:{that}\:\mathrm{cos}\:\left({n}\:\mathrm{cos}^{−\mathrm{1}} \left({a}\right)\right)\:{is}\:{also}\:{a} \\ $$$${rational}\:{number}.\:\left({n}\in\mathbb{N}\right) \\ $$

Question Number 79059 Answers: 0 Comments: 4

∫ cos^2 (x)sin^4 (x) dx ?

$$ \\ $$$$ \\ $$$$\int\:\mathrm{cos}\:^{\mathrm{2}} \left(\mathrm{x}\right)\mathrm{sin}\:^{\mathrm{4}} \left(\mathrm{x}\right)\:\mathrm{dx}\:? \\ $$

Question Number 79051 Answers: 0 Comments: 1

Is Σ_(n=1) ^∞ (1/n) converges or diverges

$${Is}\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}}\:{converges}\:{or}\:{diverges} \\ $$

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