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IntegrationQuestion and Answers: Page 217

Question Number 70369 Answers: 0 Comments: 1

Question Number 70361 Answers: 0 Comments: 0

Hello si(x)=−∫_x ^∞ ((sin(x))/x)dx show ∫_0 ^(+∞) x^(a−1) si(x)dx=−((Γ(a)sin(((πa)/2)))/a) hint ipp +complex Analysis

$${Hello}\: \\ $$$${si}\left({x}\right)=−\int_{{x}} ^{\infty} \frac{{sin}\left({x}\right)}{{x}}{dx} \\ $$$${show}\:\int_{\mathrm{0}} ^{+\infty} {x}^{{a}−\mathrm{1}} {si}\left({x}\right){dx}=−\frac{\Gamma\left({a}\right){sin}\left(\frac{\pi{a}}{\mathrm{2}}\right)}{{a}} \\ $$$${hint}\:{ipp}\:+{complex}\:{Analysis} \\ $$

Question Number 70262 Answers: 0 Comments: 1

calculate ∫_0 ^(π/4) ln(cosx)dx and ∫_0 ^(π/4) ln(sinx)dx

$${calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} {ln}\left({cosx}\right){dx}\:\:{and}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} {ln}\left({sinx}\right){dx} \\ $$

Question Number 70230 Answers: 0 Comments: 2

Question Number 70237 Answers: 0 Comments: 3

calculate ∫_0 ^∞ ((xsin(αx))/(1+x^4 ))dx with α real

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{xsin}\left(\alpha{x}\right)}{\mathrm{1}+{x}^{\mathrm{4}} }{dx}\:{with}\:\alpha\:{real} \\ $$

Question Number 70150 Answers: 0 Comments: 1

prove that ∫_0 ^(π/2) (√((4−sin^2 x)))dx < ((π(√(14)))/4)

$${prove}\:{that}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \sqrt{\left(\mathrm{4}−{sin}^{\mathrm{2}} {x}\right)}{dx}\:<\:\frac{\pi\sqrt{\mathrm{14}}}{\mathrm{4}} \\ $$

Question Number 70031 Answers: 0 Comments: 0

∫[x]dx

$$\int\left[{x}\right]{dx} \\ $$

Question Number 70718 Answers: 1 Comments: 1

∫_0 ^1 ((tan^(−1) x)/(1+x))dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{tan}^{−\mathrm{1}} \mathrm{x}}{\mathrm{1}+\mathrm{x}}\mathrm{dx} \\ $$

Question Number 69789 Answers: 0 Comments: 0

solve sin(2x)y^′ −3(cosx)y =xe^(−x)

$${solve}\:{sin}\left(\mathrm{2}{x}\right){y}^{'} \:−\mathrm{3}\left({cosx}\right){y}\:={xe}^{−{x}} \\ $$

Question Number 69784 Answers: 0 Comments: 0

calculate f(a) =∫_0 ^∞ e^(−(x^2 +(a/x^2 ))) dx with a>0

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

Question Number 69692 Answers: 0 Comments: 0

Question Number 69637 Answers: 1 Comments: 0

...now try this one: ∫(dx/(x^(1/2) −x^(1/3) −x^(1/6) ))=

$$...\mathrm{now}\:\mathrm{try}\:\mathrm{this}\:\mathrm{one}: \\ $$$$\int\frac{{dx}}{{x}^{\mathrm{1}/\mathrm{2}} −{x}^{\mathrm{1}/\mathrm{3}} −{x}^{\mathrm{1}/\mathrm{6}} }= \\ $$

Question Number 69623 Answers: 1 Comments: 0

∫(1/((√x) + (x)^(1/3) )) dx

$$\int\frac{\mathrm{1}}{\sqrt{{x}}\:+\:\sqrt[{\mathrm{3}}]{{x}}}\:{dx} \\ $$

Question Number 69603 Answers: 1 Comments: 0

∫ x^3 arcsinxdx

$$\int\:{x}^{\mathrm{3}} {arcsinxdx} \\ $$

Question Number 69597 Answers: 1 Comments: 1

Question Number 69573 Answers: 0 Comments: 1

Question Number 69570 Answers: 0 Comments: 0

Question Number 69566 Answers: 1 Comments: 0

Question Number 69565 Answers: 0 Comments: 0

Question Number 69564 Answers: 0 Comments: 3

let f(a) =∫_0 ^∞ (dx/(x^4 −2x^2 +a)) with a real and a>1 1) determine a explicit form for f(a) 2) calculate g(a) =∫_0 ^∞ (dx/((x^4 −2x^2 +a)^2 )) 3) find the values of integrals ∫_0 ^∞ (dx/(x^4 −2x^2 +3)) and ∫_0 ^∞ (dx/((x^4 −2x^2 +3)^2 ))

$${let}\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{{x}^{\mathrm{4}} −\mathrm{2}{x}^{\mathrm{2}} \:+{a}}\:\:\:{with}\:{a}\:{real}\:{and}\:{a}>\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{for}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:\:{calculate}\:{g}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{4}} −\mathrm{2}{x}^{\mathrm{2}} +{a}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{values}\:{of}\:{integrals}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{{x}^{\mathrm{4}} −\mathrm{2}{x}^{\mathrm{2}} \:+\mathrm{3}} \\ $$$${and}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{4}} −\mathrm{2}{x}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{2}} } \\ $$$$ \\ $$$$ \\ $$

Question Number 69563 Answers: 0 Comments: 1

calculate ∫_0 ^∞ (dx/(x^4 −x^2 +1))

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

Question Number 69502 Answers: 3 Comments: 2

∫((3sinx+4cosx)/(4sinx+3cosx))dx

$$\int\frac{\mathrm{3sinx}+\mathrm{4cosx}}{\mathrm{4sinx}+\mathrm{3cosx}}\mathrm{dx} \\ $$

Question Number 69390 Answers: 0 Comments: 0

study the convergence of ∫_0 ^∞ (((1+x)^α −(1+x)^((β) )/x)dx and determine its value

$${study}\:{the}\:{convergence}\:{of} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\frac{\left(\mathrm{1}+{x}\right)^{\alpha} −\left(\mathrm{1}+{x}\right)^{\left(\beta\right.} }{{x}}{dx} \\ $$$${and}\:{determine}\:{its}\:{value} \\ $$

Question Number 69389 Answers: 0 Comments: 0

find ∫_(∣z+i∣=3) ((sinz)/(z+i))dz