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IntegrationQuestion and Answers: Page 120

Question Number 126323 Answers: 0 Comments: 0

... advanced calculus... prove that ::: ((Γ(((1−x)/2))Γ(x))/(Γ((x/2)))) =^(???) ((2^(x−1) (√π))/(cos(((πx)/2))))

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\:{advanced}\:\:{calculus}... \\ $$$$\:\:\:\:\:{prove}\:\:\:{that}\:::: \\ $$$$\:\:\:\:\:\frac{\Gamma\left(\frac{\mathrm{1}−{x}}{\mathrm{2}}\right)\Gamma\left({x}\right)}{\Gamma\left(\frac{{x}}{\mathrm{2}}\right)}\:\overset{???} {=}\:\frac{\mathrm{2}^{{x}−\mathrm{1}} \sqrt{\pi}}{{cos}\left(\frac{\pi{x}}{\mathrm{2}}\right)} \\ $$$$ \\ $$

Question Number 126349 Answers: 0 Comments: 1

...nice calculus... calculate ::: Ω=^(???) ∫_0 ^( ∞) e^( −t) t^( 2) j_0 ( t )dt where : j_((v)) (x)=x^v Σ_(n=0) ^( ∞) (((−1)^n x^(2n) )/(2^(2n+v) n!Γ(n+v+1))) ::: Bessel function of the first type of order v ... j_v (x) is convergent (why?): ∀x∈R...

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{nice}\:\:{calculus}... \\ $$$$\:\:\:\:\:{calculate}\:::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Omega\overset{???} {=}\int_{\mathrm{0}} ^{\:\:\infty} {e}^{\:−{t}} \:{t}^{\:\mathrm{2}} \:{j}_{\mathrm{0}} \left(\:{t}\:\right){dt} \\ $$$$\:\:\:\:\:{where}\::\:\:{j}_{\left({v}\right)} \left({x}\right)={x}^{{v}} \underset{{n}=\mathrm{0}} {\overset{\:\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} {x}^{\mathrm{2}{n}} }{\mathrm{2}^{\mathrm{2}{n}+{v}} {n}!\Gamma\left({n}+{v}+\mathrm{1}\right)}\: \\ $$$$\:\:\:\:\:\:\:\:\::::\:{Bessel}\:{function}\:{of}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{the}\:{first}\:{type}\:{of}\:{order}\:{v}\:...\: \\ $$$$\:\:\:\:\:\:\:\:\:{j}_{{v}} \left({x}\right)\:{is}\:{convergent}\:\left({why}?\right):\:\forall{x}\in\mathbb{R}... \\ $$

Question Number 126274 Answers: 2 Comments: 0

Question Number 126273 Answers: 1 Comments: 2

Question Number 126266 Answers: 1 Comments: 0

solve ∫_0 ^( 1) ((1−x^2 )/((1+x^2 )(√(1+x^4 )))) dx ?

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

Question Number 126230 Answers: 2 Comments: 0

∫ e^( cos^(− 1) (x)) dx

$$\int\:\boldsymbol{\mathrm{e}}^{\:\boldsymbol{\mathrm{cos}}^{−\:\mathrm{1}} \left(\boldsymbol{\mathrm{x}}\right)} \:\:\:\mathrm{dx} \\ $$

Question Number 126205 Answers: 0 Comments: 0

calculate ∫∫ _([0,1]^2 ) ((dxdy)/( (√(x^2 +y^2 )) +xy))

$$\mathrm{calculate}\:\int\int\:_{\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{2}} } \:\:\:\:\:\frac{\mathrm{dxdy}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} }\:+\mathrm{xy}} \\ $$

Question Number 126203 Answers: 3 Comments: 0

∫_0 ^∞ (1/(1+x^s +x^(2s) ))

$$\overset{\infty} {\int}_{\mathrm{0}} \frac{\mathrm{1}}{\mathrm{1}+{x}^{{s}} +{x}^{\mathrm{2}{s}} } \\ $$

Question Number 126183 Answers: 1 Comments: 0

∫_0 ^( ∞) ((e^(2πx) −1)/(e^(2πx) +1)) ((1/x)−(1/(N^2 +x^2 ))) dx

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

Question Number 126179 Answers: 1 Comments: 0

let f(x)= e^(−2x) actan (3x+1) 1)calculste f^((n)) (x) and f^((n)) (0) 2) if f(x)=Σ a_n x^n determine the sequence a_n 3) calculate ∫_0 ^∞ f(x)dx

$$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)=\:\mathrm{e}^{−\mathrm{2x}} \:\mathrm{actan}\:\left(\mathrm{3x}+\mathrm{1}\right) \\ $$$$\left.\mathrm{1}\right)\mathrm{calculste}\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{x}\right)\:\mathrm{and}\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:\mathrm{if}\:\mathrm{f}\left(\mathrm{x}\right)=\Sigma\:\mathrm{a}_{\mathrm{n}} \mathrm{x}^{\mathrm{n}} \:\mathrm{determine}\:\mathrm{the}\:\mathrm{sequence}\:\mathrm{a}_{\mathrm{n}} \\ $$$$\left.\mathrm{3}\right)\:\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} \\ $$

Question Number 126139 Answers: 1 Comments: 4

closed formula .... ∫_0 ^( 1) ((x^n ln(x))/(1+x))dx =?

$$\:\:\:\:\:\:\: \\ $$$$\:\:{closed}\:\:{formula}\:.... \\ $$$$\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{x}^{{n}} {ln}\left({x}\right)}{\mathrm{1}+{x}}{dx}\:=? \\ $$$$\:\:\: \\ $$

Question Number 126133 Answers: 2 Comments: 0

Question Number 126073 Answers: 0 Comments: 2

Question Number 126068 Answers: 1 Comments: 0

Question Number 126065 Answers: 1 Comments: 0

Show that:: Ω = ∫_0 ^( 1) ((Li_2 (x)log(x))/(1+x))dx = −(3/(16))ζ(4) Goodluck

$$ \\ $$$$\mathrm{Show}\:\mathrm{that}:: \\ $$$$\:\Omega\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{Li}_{\mathrm{2}} \left(\mathrm{x}\right)\mathrm{log}\left(\mathrm{x}\right)}{\mathrm{1}+\mathrm{x}}\mathrm{dx}\:=\:−\frac{\mathrm{3}}{\mathrm{16}}\zeta\left(\mathrm{4}\right) \\ $$$$\mathrm{Goodluck} \\ $$

Question Number 126000 Answers: 3 Comments: 0

∫ (dx/((x−1)(√(x^2 −2x)))) ?

$$\:\:\:\int\:\frac{{dx}}{\left({x}−\mathrm{1}\right)\sqrt{{x}^{\mathrm{2}} −\mathrm{2}{x}}}\:? \\ $$

Question Number 125999 Answers: 2 Comments: 1

Question Number 125997 Answers: 1 Comments: 0

Find the Riemann sum for the given function with the specified number of intervals using left endpoints f(x)= 4ln x+2x ; 1≤x≤4 n=7 . Round your answer to two decimal places ?

$${Find}\:{the}\:{Riemann}\:{sum}\:{for}\:{the} \\ $$$${given}\:{function}\:{with}\:{the}\:{specified} \\ $$$${number}\:{of}\:{intervals}\:{using}\:{left} \\ $$$${endpoints}\:{f}\left({x}\right)=\:\mathrm{4ln}\:{x}+\mathrm{2}{x}\:;\:\mathrm{1}\leqslant{x}\leqslant\mathrm{4} \\ $$$${n}=\mathrm{7}\:.\:{Round}\:{your}\:{answer}\:{to}\:{two} \\ $$$${decimal}\:{places}\:? \\ $$

Question Number 125986 Answers: 1 Comments: 0