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IntegrationQuestion and Answers: Page 165

Question Number 99820 Answers: 0 Comments: 0

calculate ∫_0 ^∞ ((x^2 dx)/((x^4 −x^2 +1)^2 ))

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{x}^{\mathrm{2}} \mathrm{dx}}{\left(\mathrm{x}^{\mathrm{4}} −\mathrm{x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 99818 Answers: 2 Comments: 0

∫_0 ^1 ln(1+(1/(n^2 x^2 )))dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{ln}\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{2}} \mathrm{x}^{\mathrm{2}} }\right)\mathrm{dx} \\ $$

Question Number 99779 Answers: 1 Comments: 2

Question Number 99707 Answers: 4 Comments: 1

∫_(−∞) ^∞ e^(−x^2 ) dx=?

$$\int_{−\infty} ^{\infty} \mathrm{e}^{−\mathrm{x}^{\mathrm{2}} } \mathrm{dx}=? \\ $$

Question Number 99679 Answers: 0 Comments: 1

Question Number 99578 Answers: 2 Comments: 0

1)calculate U_n =∫_0 ^∞ e^(−nx^4 ) dx and determine lim_(n→+∞) n^4 U_n 2) find nature of the serie Σ U_n

$$\left.\mathrm{1}\right)\mathrm{calculate}\:\:\mathrm{U}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\infty} \:\mathrm{e}^{−\mathrm{nx}^{\mathrm{4}} } \mathrm{dx}\:\:\mathrm{and}\:\mathrm{determine}\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \:\mathrm{n}^{\mathrm{4}} \:\mathrm{U}_{\mathrm{n}} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{find}\:\mathrm{nature}\:\mathrm{of}\:\mathrm{the}\:\mathrm{serie}\:\Sigma\:\mathrm{U}_{\mathrm{n}} \\ $$

Question Number 99576 Answers: 2 Comments: 0

1)let f(a) =∫_0 ^∞ (( t^(a−1) ln(t))/(1+t)) dt with 0<a<1 prove that f(a)is convergent and determine it value 2)calculate∫_0 ^∞ ((lnt)/((1+t)(√t)))dt 3)calculate∫_0 ^∞ ((lnt)/((^3 (√t^2 ))(1+t)))dt

$$\left.\mathrm{1}\right)\mathrm{let}\:\mathrm{f}\left(\mathrm{a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{\:\mathrm{t}^{\mathrm{a}−\mathrm{1}} \mathrm{ln}\left(\mathrm{t}\right)}{\mathrm{1}+\mathrm{t}}\:\mathrm{dt}\:\:\:\mathrm{with}\:\mathrm{0}<\mathrm{a}<\mathrm{1}\:\:\:\mathrm{prove}\:\mathrm{that}\:\mathrm{f}\left(\mathrm{a}\right)\mathrm{is}\:\mathrm{convergent}\:\mathrm{and}\:\mathrm{determine} \\ $$$$\mathrm{it}\:\mathrm{value} \\ $$$$\left.\mathrm{2}\right)\mathrm{calculate}\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{lnt}}{\left(\mathrm{1}+\mathrm{t}\right)\sqrt{\mathrm{t}}}\mathrm{dt} \\ $$$$\left.\mathrm{3}\right)\mathrm{calculate}\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{lnt}}{\left(^{\mathrm{3}} \sqrt{\mathrm{t}^{\mathrm{2}} }\right)\left(\mathrm{1}+\mathrm{t}\right)}\mathrm{dt} \\ $$

Question Number 99557 Answers: 1 Comments: 0

Λ=∫_0 ^1 (Li_2 ((1/(1+x)))+Li_3 ((x/(1+x)))+Li_4 (((x+1)/(x^2 +x+1))))dx Li_n (z)=polylogarithm function. by adeyemi.

$$ \\ $$$$\Lambda=\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{Li}_{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{1}+\mathrm{x}}\right)+\mathrm{Li}_{\mathrm{3}} \left(\frac{\mathrm{x}}{\mathrm{1}+\mathrm{x}}\right)+\mathrm{Li}_{\mathrm{4}} \left(\frac{\mathrm{x}+\mathrm{1}}{\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}}\right)\right)\mathrm{dx} \\ $$$$\mathrm{Li}_{\mathrm{n}} \left(\mathrm{z}\right)=\mathrm{polylogarithm}\:\mathrm{function}. \\ $$$$\mathrm{by}\:\mathrm{adeyemi}. \\ $$$$ \\ $$

Question Number 99541 Answers: 3 Comments: 0

∫_(−∞) ^∞ e^(−2x^2 −5x−3) dx=? help me

$$\int_{−\infty} ^{\infty} \mathrm{e}^{−\mathrm{2x}^{\mathrm{2}} −\mathrm{5x}−\mathrm{3}} \mathrm{dx}=?\: \\ $$$$\mathrm{help}\:\mathrm{me} \\ $$

Question Number 99536 Answers: 0 Comments: 0

∫tan^(πi) xdx

$$\int{tan}^{\pi{i}} {xdx} \\ $$

Question Number 99516 Answers: 1 Comments: 0

Question Number 99504 Answers: 4 Comments: 0

Question Number 99496 Answers: 1 Comments: 0

convergence radius of Σ_(n∈N) 2^n z^(n!)

$${convergence}\:{radius}\:{of}\:\:\underset{{n}\in\mathbb{N}} {\sum}\:\mathrm{2}^{{n}} {z}^{{n}!} \: \\ $$

Question Number 99421 Answers: 0 Comments: 0

Question Number 99413 Answers: 1 Comments: 0

∫_0 ^(+∞) ((sin(ax))/(e^(2πx) −1))dx

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

Question Number 99403 Answers: 1 Comments: 1

∫ x^x dx

$$\int\:\mathrm{x}^{\mathrm{x}} \:\:\mathrm{dx} \\ $$

Question Number 99326 Answers: 1 Comments: 0

Question Number 99315 Answers: 0 Comments: 0

Question Number 99278 Answers: 0 Comments: 0

x 0 2 4 6 8 10 f(x) 2.4 3.6 4.9 6.9 8.0 11.9 Given the curve y = f(x), with corresponding values of f(x) at certain x values. The curve y = f(x) is rotated at an angle of 2π about the x−axis. Find (1) The area of the surface generated using simpson′s rule. (2) The volume of the surface generated using simpson′s rule.

$$\:{x}\:\:\:\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}\:\:\:\:\:\:\:\:\:\mathrm{4}\:\:\:\:\:\:\:\:\:\:\mathrm{6}\:\:\:\:\:\:\:\:\:\mathrm{8}\:\:\:\:\:\:\:\:\:\:\mathrm{10} \\ $$$$\:{f}\left({x}\right)\:\:\mathrm{2}.\mathrm{4}\:\:\:\:\:\:\:\mathrm{3}.\mathrm{6}\:\:\:\:\:\:\mathrm{4}.\mathrm{9}\:\:\:\:\:\mathrm{6}.\mathrm{9}\:\:\:\:\:\mathrm{8}.\mathrm{0}\:\:\:\:\:\:\:\:\mathrm{11}.\mathrm{9} \\ $$$$\mathrm{Given}\:\mathrm{the}\:\mathrm{curve}\:{y}\:=\:{f}\left({x}\right),\:\mathrm{with}\:\mathrm{corresponding}\:\mathrm{values}\:\mathrm{of} \\ $$$${f}\left({x}\right)\:\mathrm{at}\:\mathrm{certain}\:{x}\:\mathrm{values}.\:\mathrm{The}\:\mathrm{curve}\:{y}\:=\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{rotated} \\ $$$$\mathrm{at}\:\mathrm{an}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{2}\pi\:\mathrm{about}\:\mathrm{the}\:\mathrm{x}−\mathrm{axis}.\:\mathrm{Find}\: \\ $$$$\left(\mathrm{1}\right)\:\mathrm{The}\:\mathrm{area}\:\mathrm{of}\:\mathrm{the}\:\mathrm{surface}\:\mathrm{generated}\:\mathrm{using}\:\mathrm{simpson}'\mathrm{s}\:\mathrm{rule}. \\ $$$$\left(\mathrm{2}\right)\:\mathrm{The}\:\mathrm{volume}\:\mathrm{of}\:\mathrm{the}\:\mathrm{surface}\:\mathrm{generated}\:\mathrm{using}\:\mathrm{simpson}'\mathrm{s}\:\mathrm{rule}. \\ $$

Question Number 99261 Answers: 1 Comments: 0

Given F(x)=(1/2)∫((x+1)/(x−1))f(t)dt Show that F is defined, continuous, and derivable. And find its derivative

$$\mathrm{Given}\:\mathrm{F}\left(\mathrm{x}\right)=\frac{\mathrm{1}}{\mathrm{2}}\int\frac{\mathrm{x}+\mathrm{1}}{\mathrm{x}−\mathrm{1}}\mathrm{f}\left(\mathrm{t}\right)\mathrm{dt} \\ $$$$\mathrm{Show}\:\mathrm{that}\:\mathrm{F}\:\mathrm{is}\:\mathrm{defined},\:\mathrm{continuous},\:\mathrm{and}\:\mathrm{derivable}. \\ $$$$\mathrm{And}\:\mathrm{find}\:\mathrm{its}\:\mathrm{derivative} \\ $$

Question Number 99239 Answers: 2 Comments: 0

calculate ∫_2 ^(+∞) (dx/(x^3 (x^2 −1)^2 ))

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

Question Number 99234 Answers: 1 Comments: 0

calculate ∫_0 ^∞ e^(−x) lnx dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\mathrm{e}^{−\mathrm{x}} \mathrm{lnx}\:\mathrm{dx} \\ $$

Question Number 99228 Answers: 1 Comments: 0

∫_0 ^∞ ((sin(x)ln(x))/x)dx=((−γπ)/2)