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IntegrationQuestion and Answers: Page 186

Question Number 88859 Answers: 0 Comments: 1

∫ e^(ax) cos bx dx ∫x^2 e^(2x) ln3x^2 dx

$$\int\:\boldsymbol{{e}}^{\boldsymbol{{ax}}} \mathrm{cos}\:\boldsymbol{{bx}}\:\boldsymbol{{dx}} \\ $$$$\int\boldsymbol{{x}}^{\mathrm{2}} \boldsymbol{{e}}^{\mathrm{2}\boldsymbol{{x}}} \boldsymbol{{ln}}\mathrm{3}\boldsymbol{{x}}^{\mathrm{2}} \:\boldsymbol{{dx}} \\ $$

Question Number 88852 Answers: 1 Comments: 0

prove that ∫_0 ^n ⌈x⌉dx= ((n(n+1))/2) and ∫_0 ^n ⌊x⌋dx=((n(n−1))/2) when ⌊..⌋ is floor and ⌈..⌉ is ceil

$${prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{{n}} \lceil{x}\rceil{dx}=\:\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}\:{and}\:\int_{\mathrm{0}} ^{{n}} \lfloor{x}\rfloor{dx}=\frac{{n}\left({n}−\mathrm{1}\right)}{\mathrm{2}} \\ $$$${when}\:\lfloor..\rfloor\:{is}\:{floor}\:{and}\:\lceil..\rceil\:{is}\:{ceil} \\ $$

Question Number 88789 Answers: 0 Comments: 7

∫∫ln(x+1) dx dy

$$\int\int{ln}\left({x}+\mathrm{1}\right)\:{dx}\:{dy} \\ $$

Question Number 88710 Answers: 2 Comments: 1

∫_(−(√3) ) ^(√3) ∫_1 ^(√(4−x^2 )) (x^2 +y^2 )^(3/2) dydx

$$\underset{−\sqrt{\mathrm{3}}\:} {\overset{\sqrt{\mathrm{3}}} {\int}}\underset{\mathrm{1}} {\overset{\sqrt{\mathrm{4}−{x}^{\mathrm{2}} }} {\int}}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)^{\mathrm{3}/\mathrm{2}} \:{dydx} \\ $$

Question Number 88616 Answers: 1 Comments: 9

Question Number 88606 Answers: 2 Comments: 2

Question Number 88603 Answers: 1 Comments: 0

Question Number 88586 Answers: 1 Comments: 0

∫((√(cos(2x)+3))/(cos(x)))dx

$$\int\frac{\sqrt{{cos}\left(\mathrm{2}{x}\right)+\mathrm{3}}}{{cos}\left({x}\right)}{dx} \\ $$

Question Number 88569 Answers: 1 Comments: 1

Question Number 88555 Answers: 1 Comments: 0

slove ⌈(x/a)⌉<a when a>1 ⌈...⌉ is ceil function

$${slove}\: \\ $$$$\lceil\frac{{x}}{{a}}\rceil<{a}\:\:\: \\ $$$${when}\:{a}>\mathrm{1} \\ $$$$\lceil...\rceil\:{is}\:{ceil}\:{function} \\ $$

Question Number 88547 Answers: 1 Comments: 0

prove for (0<a<2) ∫_0 ^( ∞) ((x^(a−1) dx)/(1+x+x^2 )) = ((2π)/(√3))cos (((2πa+π)/6))cosec πa .

$${prove}\:{for}\:\left(\mathrm{0}<{a}<\mathrm{2}\right) \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \frac{{x}^{{a}−\mathrm{1}} {dx}}{\mathrm{1}+{x}+{x}^{\mathrm{2}} }\:=\:\frac{\mathrm{2}\pi}{\sqrt{\mathrm{3}}}\mathrm{cos}\:\left(\frac{\mathrm{2}\pi{a}+\pi}{\mathrm{6}}\right)\mathrm{cosec}\:\pi{a}\:. \\ $$

Question Number 88490 Answers: 0 Comments: 4

∫_1 ^∞ (x^4 /4^x )dx=?

$$\int_{\mathrm{1}} ^{\infty} \:\frac{{x}^{\mathrm{4}} }{\mathrm{4}^{{x}} }{dx}=? \\ $$

Question Number 88462 Answers: 0 Comments: 3

Question Number 88438 Answers: 2 Comments: 0

∫((x^5 +1)/(x^5 −1))dx

$$\int\frac{{x}^{\mathrm{5}} +\mathrm{1}}{{x}^{\mathrm{5}} −\mathrm{1}}{dx} \\ $$

Question Number 88429 Answers: 0 Comments: 0

show that ∫_(0 ) ^1 ln(x) sin^(−1) (√x) dx= (π/2)(ln(2)−1)

$${show}\:{that} \\ $$$$\int_{\mathrm{0}\:} ^{\mathrm{1}} {ln}\left({x}\right)\:{sin}^{−\mathrm{1}} \sqrt{{x}}\:{dx}=\:\frac{\pi}{\mathrm{2}}\left({ln}\left(\mathrm{2}\right)−\mathrm{1}\right) \\ $$

Question Number 88422 Answers: 0 Comments: 1

calculate ∫_1 ^∞ (dx/((x+1)^3 (x^2 +1)^2 ))

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

Question Number 88415 Answers: 0 Comments: 2

find L(((1−cosx)/x^2 )) with L lsplace transform

$${find}\:{L}\left(\frac{\mathrm{1}−{cosx}}{{x}^{\mathrm{2}} }\right)\:{with}\:{L}\:{lsplace}\:{transform} \\ $$

Question Number 88414 Answers: 0 Comments: 2

find approcimstive value of ∫_(π/3) ^(π/2) (x/(sinx))dx

$${find}\:{approcimstive}\:{value}\:{of}\:\:\:\int_{\frac{\pi}{\mathrm{3}}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{{x}}{{sinx}}{dx} \\ $$

Question Number 88413 Answers: 0 Comments: 3

∫_0 ^∞ e^(−x^2 ) dx

$$\int_{\mathrm{0}} ^{\infty} \:\mathrm{e}^{−\mathrm{x}^{\mathrm{2}} } \mathrm{dx} \\ $$$$ \\ $$

Question Number 88388 Answers: 1 Comments: 0

Question Number 88357 Answers: 0 Comments: 1

∫ _1 ^4 (dx/((4x−1)(√x)))

$$\int\underset{\mathrm{1}} {\overset{\mathrm{4}} {\:}}\:\frac{\mathrm{dx}}{\left(\mathrm{4x}−\mathrm{1}\right)\sqrt{\mathrm{x}}} \\ $$

Question Number 88307 Answers: 1 Comments: 0

∫(x^2 /(x^2 −(5/2)x−(3/2))) dx

$$\int\frac{{x}^{\mathrm{2}} }{{x}^{\mathrm{2}} −\frac{\mathrm{5}}{\mathrm{2}}{x}−\frac{\mathrm{3}}{\mathrm{2}}}\:{dx} \\ $$

Question Number 88286 Answers: 0 Comments: 1

Question Number 88253 Answers: 0 Comments: 0

∫ ((ln(x^2 +1) dx)/(x+1))

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

Question Number 88206 Answers: 1 Comments: 0

∫ ((x+x^3 )/(1+x^4 )) dx

$$\int\:\:\frac{\mathrm{x}+\mathrm{x}^{\mathrm{3}} }{\mathrm{1}+\mathrm{x}^{\mathrm{4}} }\:\mathrm{dx}\: \\ $$

Question Number 88197 Answers: 0 Comments: 0

prove that ∫_0 ^1 (√(((x^2 −2)/(x^2 −1)) ))dx=((π(√(2π)))/(Γ^2 ((1/4))))+((Γ^2 ((1/4)))/(4(√(2π))))

$${prove}\:{that}\: \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\frac{{x}^{\mathrm{2}} −\mathrm{2}}{{x}^{\mathrm{2}} −\mathrm{1}}\:}{dx}=\frac{\pi\sqrt{\mathrm{2}\pi}}{\Gamma^{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{4}}\right)}+\frac{\Gamma^{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{4}}\right)}{\mathrm{4}\sqrt{\mathrm{2}\pi}} \\ $$

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