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IntegrationQuestion and Answers: Page 201

Question Number 80334 Answers: 0 Comments: 1

let f∈L^1 (R) let u_n = ∫_a ^b f(t)sin(nt)dt , v_n =∫_a ^b ((f(t))/t)sin(nt) 1)Prove that lim_(n→∞) u_n =0 2)Deduce in term of a,b,f(0) the value of lim_(n→∞) v_n

$$\:{let}\:\:\:{f}\in{L}^{\mathrm{1}} \left(\mathbb{R}\right)\:\:\: \\ $$$${let}\:\:{u}_{{n}} =\:\int_{{a}} ^{{b}} {f}\left({t}\right){sin}\left({nt}\right){dt}\:,\:{v}_{{n}} =\int_{{a}} ^{{b}} \frac{{f}\left({t}\right)}{{t}}{sin}\left({nt}\right)\: \\ $$$$\left.\mathrm{1}\right){Prove}\:{that}\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:{u}_{{n}} =\mathrm{0} \\ $$$$\left.\mathrm{2}\right){Deduce}\:\:{in}\:{term}\:{of}\:{a},{b},{f}\left(\mathrm{0}\right)\:{the}\:{value}\:{of}\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:{v}_{{n}} \:\: \\ $$

Question Number 80332 Answers: 0 Comments: 1

let α ∈R and a_n =Σ_(k=1) ^n ((sin(kα))/(n+k)) Find lim_(n→∞) a_n

$$\:\:{let}\:\alpha\:\in\mathbb{R}\:\:{and}\:\:\:\:{a}_{{n}} =\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{{sin}\left({k}\alpha\right)}{{n}+{k}} \\ $$$${Find}\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\:{a}_{{n}} \: \\ $$

Question Number 80312 Answers: 1 Comments: 17

Question Number 80300 Answers: 0 Comments: 4

Question Number 80227 Answers: 0 Comments: 5

how to prove ∫_0 ^1 x^n (1−x)^(m ) dx = ((m! ×n!)/((m+n)!)) via Gamma function

$${how}\:{to}\:{prove} \\ $$$$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:{x}^{{n}} \:\left(\mathrm{1}−{x}\right)^{{m}\:} \:{dx}\:=\:\frac{{m}!\:×{n}!}{\left({m}+{n}\right)!} \\ $$$${via}\:{Gamma}\:{function} \\ $$

Question Number 80052 Answers: 0 Comments: 0

∫ e^(sin 2x) .cos x dx =

$$\int\:\mathrm{e}^{\mathrm{sin}\:\mathrm{2x}} .\mathrm{cos}\:\mathrm{x}\:\mathrm{dx}\:= \\ $$$$ \\ $$

Question Number 79929 Answers: 0 Comments: 0

∫e^(√(sin x)) dx=?

$$\int{e}^{\sqrt{\mathrm{sin}\:{x}}} {dx}=? \\ $$

Question Number 79913 Answers: 0 Comments: 1

Convergence of I=∫_0 ^( ∞) (e^t /(e^(−t) +e^(2t) ∣sint∣))dt

$$\:{Convergence}\:\:{of}\:\:{I}=\int_{\mathrm{0}} ^{\:\infty} \frac{{e}^{{t}} }{{e}^{−{t}} +{e}^{\mathrm{2}{t}} \mid{sint}\mid}{dt} \\ $$

Question Number 79903 Answers: 1 Comments: 11

Question Number 79869 Answers: 0 Comments: 1

For witch value of α the integral I=∫_0 ^∞ ((1/(√(1+2x^2 )))−(α/(1+x)))dx converge; and in this case calculate α

$${For}\:\:{witch}\:\:{value}\:\:{of}\:\:\alpha\:\:{the}\:\:{integral} \\ $$$$\:\:{I}=\int_{\mathrm{0}} ^{\infty} \left(\frac{\mathrm{1}}{\sqrt{\mathrm{1}+\mathrm{2}{x}^{\mathrm{2}} }}−\frac{\alpha}{\mathrm{1}+{x}}\right){dx}\:\:{converge}; \\ $$$$\:\:{and}\:\:{in}\:\:{this}\:\:{case}\:\:{calculate}\:\:\alpha \\ $$

Question Number 79837 Answers: 1 Comments: 10

Question Number 79825 Answers: 0 Comments: 4

∫_( 0) ^( 1) (√(x^3 + 1)) dx

$$\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\sqrt{\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{1}}\:\:\mathrm{dx} \\ $$

Question Number 79824 Answers: 2 Comments: 7

Question Number 79814 Answers: 0 Comments: 5

Question Number 79763 Answers: 1 Comments: 2

calculate ∫_0 ^π {cos^8 x +sin^8 x}dx

$${calculate}\:\int_{\mathrm{0}} ^{\pi} \left\{{cos}^{\mathrm{8}} {x}\:+{sin}^{\mathrm{8}} {x}\right\}{dx} \\ $$

Question Number 79758 Answers: 0 Comments: 1

find value of ∫_0 ^1 ln(1+ix^2 )dx and ∫_0 ^1 ln(1−ix^2 )dx with i=(√(−1))

$${find}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+{ix}^{\mathrm{2}} \right){dx}\:{and}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}−{ix}^{\mathrm{2}} \right){dx}\:{with}\:{i}=\sqrt{−\mathrm{1}} \\ $$

Question Number 79730 Answers: 1 Comments: 1

I) For witch value of α the integral C=∫_0 ^( ∞) ((1/(√(1+2x^2 )))−(1/(x+1)))dx conveege ? And in this case calculate α. II) Let Δ={(x; y)/ ∣x∣+∣y∣≤2} a) Calculate I_1 = ∫∫_Δ dxdy and ∫∫_Δ ((dxdy)/((∣x∣+∣y∣)^2 +4))

$$\left.{I}\right)\:\:{For}\:{witch}\:{value}\:{of}\:\alpha\:{the}\:{integral} \\ $$$$\:{C}=\int_{\mathrm{0}} ^{\:\infty} \left(\frac{\mathrm{1}}{\sqrt{\mathrm{1}+\mathrm{2}{x}^{\mathrm{2}} }}−\frac{\mathrm{1}}{{x}+\mathrm{1}}\right){dx}\:\:{conveege}\:\:? \\ $$$${And}\:{in}\:{this}\:{case}\:{calculate}\:\alpha. \\ $$$$\left.{II}\right)\:\:{Let}\:\Delta=\left\{\left({x};\:{y}\right)/\:\mid{x}\mid+\mid{y}\mid\leqslant\mathrm{2}\right\} \\ $$$$\left.\:\:\:\:\:{a}\right)\:{Calculate}\:{I}_{\mathrm{1}} =\:\int\int_{\Delta} {dxdy}\:\:\:{and}\:\:\int\int_{\Delta} \frac{{dxdy}}{\left(\mid{x}\mid+\mid{y}\mid\right)^{\mathrm{2}} +\mathrm{4}} \\ $$

Question Number 79646 Answers: 0 Comments: 1

calculate A_n =∫_0 ^1 cos(narcosx)dx with n integr natural

$${calculate}\:\:{A}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:{cos}\left({narcosx}\right){dx} \\ $$$${with}\:{n}\:{integr}\:{natural} \\ $$

Question Number 79645 Answers: 0 Comments: 0

find ∫_0 ^1 ln(1+x^4 )dx

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{ln}\left(\mathrm{1}+{x}^{\mathrm{4}} \right){dx} \\ $$

Question Number 79634 Answers: 1 Comments: 3