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IntegrationQuestion and Answers: Page 163

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)

$$ \\ $$$$\int_{\mathrm{0}} ^{\infty} \frac{{sin}\left({x}\right){ln}\left({x}\right)}{{x}}{dx}=\frac{−\gamma\pi}{\mathrm{2}} \\ $$

Question Number 99205 Answers: 2 Comments: 1

Question Number 99168 Answers: 1 Comments: 0

Question Number 99154 Answers: 4 Comments: 0

Question Number 99146 Answers: 1 Comments: 0

1) explicit f(a) =∫_1 ^(√3) arctan((a/x))dx with a>0 2) calculate ∫_1 ^(√3) arctan((2/x))dx and ∫_1 ^(√3) arctan((3/x))dx

$$\left.\mathrm{1}\right)\:\mathrm{explicit}\:\mathrm{f}\left(\mathrm{a}\right)\:=\int_{\mathrm{1}} ^{\sqrt{\mathrm{3}}} \:\:\mathrm{arctan}\left(\frac{\mathrm{a}}{\mathrm{x}}\right)\mathrm{dx}\:\:\mathrm{with}\:\mathrm{a}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{calculate}\:\int_{\mathrm{1}} ^{\sqrt{\mathrm{3}}} \:\mathrm{arctan}\left(\frac{\mathrm{2}}{\mathrm{x}}\right)\mathrm{dx}\:\mathrm{and}\:\int_{\mathrm{1}} ^{\sqrt{\mathrm{3}}} \:\:\mathrm{arctan}\left(\frac{\mathrm{3}}{\mathrm{x}}\right)\mathrm{dx} \\ $$

Question Number 99120 Answers: 0 Comments: 2

prove that: ∫_(−(1/2)) ^∞ e^(−(4x^6 +12x^5 +15x^4 +10x^3 +4x^2 +x)) dx =((e)^(1/8) /3)[((Γ((1/6))^((−1)/2) )/(2(2)^(1/3) ))1F2(_(1/3,2/3) ^(1/6) ∣((−1)/(69/2)) ) +((Γ(5/6))/(128(4)^(1/3) ))1F2(_(4/3,5/3) ^(5/6) ∣((−1)/(69/2))) −((√π)/(16))12(_(2/3,4/3) ^(1/2) ∣((−1)/(69/2)))

$${prove}\:{that}: \\ $$$$\int_{−\frac{\mathrm{1}}{\mathrm{2}}} ^{\infty} {e}^{−\left(\mathrm{4}{x}^{\mathrm{6}} +\mathrm{12}{x}^{\mathrm{5}} +\mathrm{15}{x}^{\mathrm{4}} +\mathrm{10}{x}^{\mathrm{3}} +\mathrm{4}{x}^{\mathrm{2}} +{x}\right)} {dx} \\ $$$$=\frac{\sqrt[{\mathrm{8}}]{{e}}}{\mathrm{3}}\left[\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{6}}\right)^{\frac{−\mathrm{1}}{\mathrm{2}}} }{\mathrm{2}\sqrt[{\mathrm{3}}]{\mathrm{2}}}\mathrm{1}{F}\mathrm{2}\left(_{\mathrm{1}/\mathrm{3},\mathrm{2}/\mathrm{3}} ^{\mathrm{1}/\mathrm{6}} \mid\frac{−\mathrm{1}}{\mathrm{69}/\mathrm{2}}\:\right)\:+\frac{\Gamma\left(\mathrm{5}/\mathrm{6}\right)}{\mathrm{128}\sqrt[{\mathrm{3}}]{\mathrm{4}}}\mathrm{1}{F}\mathrm{2}\left(_{\mathrm{4}/\mathrm{3},\mathrm{5}/\mathrm{3}} ^{\mathrm{5}/\mathrm{6}} \mid\frac{−\mathrm{1}}{\mathrm{69}/\mathrm{2}}\right)\:−\frac{\sqrt{\pi}}{\mathrm{16}}\mathrm{12}\left(_{\mathrm{2}/\mathrm{3},\mathrm{4}/\mathrm{3}} ^{\mathrm{1}/\mathrm{2}} \mid\frac{−\mathrm{1}}{\mathrm{69}/\mathrm{2}}\right)\:\right. \\ $$

Question Number 99114 Answers: 1 Comments: 0

calculate: ∫(√x)sinh^(−1) (x)dx where sinh^(−1) (x) is the inverse hyperbolic sine function

$${calculate}: \\ $$$$\int\sqrt{{x}}{sinh}^{−\mathrm{1}} \left({x}\right){dx} \\ $$$${where}\:{sinh}^{−\mathrm{1}} \left({x}\right)\:{is}\:{the}\:{inverse}\:{hyperbolic}\: \\ $$$${sine}\:{function} \\ $$$$ \\ $$$$ \\ $$

Question Number 99044 Answers: 3 Comments: 0

∫(1/(x^2 +1))dx=?

$$\int\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} +\mathrm{1}}\mathrm{dx}=? \\ $$

Question Number 99007 Answers: 2 Comments: 0

Let I_y = ∫_(−2) ^2 [y^3 cos ((y/2)) + (1/2)]((√(4−y^2 )) ) dy then I_y = ???

$$\mathrm{Let}\:{I}_{{y}} \:=\:\underset{−\mathrm{2}} {\overset{\mathrm{2}} {\int}}\left[{y}^{\mathrm{3}} \:\mathrm{cos}\:\left(\frac{{y}}{\mathrm{2}}\right)\:+\:\frac{\mathrm{1}}{\mathrm{2}}\right]\left(\sqrt{\mathrm{4}−{y}^{\mathrm{2}} }\:\right)\:{dy}\: \\ $$$$\mathrm{then}\:{I}_{{y}} \:=\:??? \\ $$

Question Number 98951 Answers: 0 Comments: 1

∫tan^(1/5) x.cotx.secxdx