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IntegrationQuestion and Answers: Page 213

Question Number 74889 Answers: 1 Comments: 1

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

$${find}\:\int_{−\frac{\mathrm{1}}{\mathrm{2}}} ^{+\infty} \:\:{e}^{−{x}} \sqrt{\mathrm{2}{x}+\mathrm{1}}{dx} \\ $$

Question Number 74888 Answers: 1 Comments: 3

calculate f(α)=∫(√(x^2 −x+α))dx (α real)

$${calculate}\:{f}\left(\alpha\right)=\int\sqrt{{x}^{\mathrm{2}} −{x}+\alpha}{dx}\:\:\left(\alpha\:{real}\right) \\ $$

Question Number 74886 Answers: 0 Comments: 1

calculate ∫ ((x+1)/((x^3 +x−2)^2 ))dx

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

Question Number 74861 Answers: 1 Comments: 0

Question Number 74799 Answers: 1 Comments: 0

prove that 0≤∫_0 ^∞ ((t^2 e^(−nt) )/(e^t −1))dt ≤(1/n^2 ) for n integr not 0

$${prove}\:{that}\:\mathrm{0}\leqslant\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{t}^{\mathrm{2}} \:{e}^{−{nt}} }{{e}^{{t}} −\mathrm{1}}{dt}\:\leqslant\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\:\:{for}\:{n}\:{integr}\:{not}\:\mathrm{0} \\ $$

Question Number 74798 Answers: 0 Comments: 0

calculate ∫_0 ^π ln(1−2xcosθ +x^2 )dθ with ∣x∣<1

$${calculate}\:\int_{\mathrm{0}} ^{\pi} {ln}\left(\mathrm{1}−\mathrm{2}{xcos}\theta\:+{x}^{\mathrm{2}} \right){d}\theta\:\:{with}\:\mid{x}\mid<\mathrm{1} \\ $$

Question Number 74796 Answers: 0 Comments: 0

let U_n =(−1)^n {arcsin((1/n))−(1/n)}^(1/3) study the convergence of Σ U_n

$${let}\:{U}_{{n}} =\left(−\mathrm{1}\right)^{{n}} \left\{{arcsin}\left(\frac{\mathrm{1}}{{n}}\right)−\frac{\mathrm{1}}{{n}}\right\}^{\frac{\mathrm{1}}{\mathrm{3}}} \\ $$$${study}\:{the}\:{convergence}\:{of}\:\Sigma\:{U}_{{n}} \\ $$

Question Number 74793 Answers: 1 Comments: 1

prove the convergence of ∫_0 ^1 ((ln(1+(√x)))/(√x))dx

$${prove}\:{the}\:{convergence}\:{of}\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}+\sqrt{{x}}\right)}{\sqrt{{x}}}{dx} \\ $$

Question Number 74720 Answers: 2 Comments: 5

Question Number 74944 Answers: 0 Comments: 0

∫(e^(−cos(2x)) /(sin^2 (x))) dx

$$\int\frac{{e}^{−{cos}\left(\mathrm{2}{x}\right)} }{{sin}^{\mathrm{2}} \left({x}\right)}\:{dx} \\ $$

Question Number 74621 Answers: 1 Comments: 1

Question Number 74620 Answers: 1 Comments: 0

Question Number 74514 Answers: 1 Comments: 1

calculate ∫_0 ^(2π) (((x−sinθ)dθ)/((x^2 −2x sinθ +1)^2 ))

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{\left({x}−{sin}\theta\right){d}\theta}{\left({x}^{\mathrm{2}} −\mathrm{2}{x}\:{sin}\theta\:+\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 74500 Answers: 0 Comments: 0

1) decompose the fraction F(x)=((x^3 −2)/((x+1)^4 (x−2)^3 )) 2)find ∫_3 ^(+∞) F(x)dx

$$\left.\mathrm{1}\right)\:{decompose}\:{the}\:{fraction}\:{F}\left({x}\right)=\frac{{x}^{\mathrm{3}} −\mathrm{2}}{\left({x}+\mathrm{1}\right)^{\mathrm{4}} \left({x}−\mathrm{2}\right)^{\mathrm{3}} } \\ $$$$\left.\mathrm{2}\right){find}\:\:\:\int_{\mathrm{3}} ^{+\infty} \:{F}\left({x}\right){dx} \\ $$

Question Number 74492 Answers: 0 Comments: 0

calculate ∫_0 ^∞ ((arctan(x^2 ))/(x^2 +9))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left({x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} \:+\mathrm{9}}{dx} \\ $$

Question Number 74484 Answers: 0 Comments: 1

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

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

Question Number 74446 Answers: 1 Comments: 0

Question Number 75089 Answers: 0 Comments: 4

∫sin (x^3 +c) dx=? ∫sinh (x^3 +c) dx=?

$$\int\mathrm{sin}\:\left({x}^{\mathrm{3}} +{c}\right)\:{dx}=? \\ $$$$\int\mathrm{sinh}\:\left({x}^{\mathrm{3}} +{c}\right)\:{dx}=? \\ $$

Question Number 74394 Answers: 1 Comments: 4

Question Number 74367 Answers: 0 Comments: 2

Solve : (D^4 +4)y=0 given: y(0)=0 , y′(0)=2 , y′′(0)=0 and y′′′(0)=4.

$${Solve}\:: \\ $$$$\left({D}^{\mathrm{4}} +\mathrm{4}\right){y}=\mathrm{0}\: \\ $$$${given}:\:{y}\left(\mathrm{0}\right)=\mathrm{0}\:,\:{y}'\left(\mathrm{0}\right)=\mathrm{2}\:,\:{y}''\left(\mathrm{0}\right)=\mathrm{0}\:{and} \\ $$$${y}'''\left(\mathrm{0}\right)=\mathrm{4}. \\ $$

Question Number 74395 Answers: 0 Comments: 2

calculate ∫_0 ^∞ e^(−2x) [e^x ]dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:{e}^{−\mathrm{2}{x}} \left[{e}^{{x}} \right]{dx} \\ $$

Question Number 74349 Answers: 0 Comments: 2

calculate ∫_0 ^∞ ((cos(2πx))/((x^2 +3)^2 ))dx

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

Question Number 74348 Answers: 0 Comments: 1

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

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

Question Number 74347 Answers: 0 Comments: 0

calculate ∫_0 ^∞ ((arctan(cos(πx^2 )))/(x^2 +1))dx

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

Question Number 74346 Answers: 1 Comments: 0

find ∫ ((x+(√(x+1)))/(2(√(x−1))+3))dx

$${find}\:\int\:\:\frac{{x}+\sqrt{{x}+\mathrm{1}}}{\mathrm{2}\sqrt{{x}−\mathrm{1}}+\mathrm{3}}{dx} \\ $$

Question Number 74344 Answers: 1 Comments: 1

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

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

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