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IntegrationQuestion and Answers: Page 267

Question Number 41518 Answers: 3 Comments: 0

calculate A_n = ∫_0 ^1 (1−t^2 )^n dt with n integr natural

$${calculate}\:\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{t}^{\mathrm{2}} \right)^{{n}} {dt}\:\:\:{with}\:{n}\:{integr}\:{natural} \\ $$

Question Number 41516 Answers: 1 Comments: 1

calculate ∫_0 ^1 ((ln(1+x))/((1+x)^4 )) dx

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

Question Number 41515 Answers: 1 Comments: 3

let f_n (x) =((sin(2(n+1)x))/(sinx)) if x∈]0,(π/2)] and f_n (0)=2(n+1) let u_n = ∫_0 ^(π/2) f_n (x)dx 1) prove that ∀n fromN u_(n+1) −u_n =2(((−1)^(n+1) )/(2n+3)) 2)find lim_(n→+∞) u_n

$$\left.{l}\left.{et}\:\:{f}_{{n}} \left({x}\right)\:=\frac{{sin}\left(\mathrm{2}\left({n}+\mathrm{1}\right){x}\right)}{{sinx}}\:{if}\:\:{x}\in\right]\mathrm{0},\frac{\pi}{\mathrm{2}}\right]\:{and}\:{f}_{{n}} \left(\mathrm{0}\right)=\mathrm{2}\left({n}+\mathrm{1}\right)\:\:{let} \\ $$$${u}_{{n}} =\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{f}_{{n}} \left({x}\right){dx} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\forall{n}\:{fromN}\:\:{u}_{{n}+\mathrm{1}} −{u}_{{n}} =\mathrm{2}\frac{\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} }{\mathrm{2}{n}+\mathrm{3}} \\ $$$$\left.\mathrm{2}\right){find}\:{lim}_{{n}\rightarrow+\infty} \:{u}_{{n}} \\ $$

Question Number 41514 Answers: 1 Comments: 0

find ∫ cos(lnx)dx

$${find}\:\:\:\int\:\:{cos}\left({lnx}\right){dx}\: \\ $$

Question Number 41487 Answers: 3 Comments: 4

Question Number 41461 Answers: 1 Comments: 3

Find area of square inserted in curve f(x)= 3x−x^3 .

$$\mathrm{Find}\:\mathrm{area}\:\mathrm{of}\:\mathrm{square}\:\mathrm{inserted}\:\mathrm{in}\:\mathrm{curve} \\ $$$$\mathrm{f}\left({x}\right)=\:\mathrm{3}{x}−{x}^{\mathrm{3}} . \\ $$

Question Number 41436 Answers: 4 Comments: 0

Question Number 41378 Answers: 1 Comments: 1

Solve : e^x (x+1)dx + (ye^y − xe^x )dy=0

$$\mathrm{Solve}\:: \\ $$$$\mathrm{e}^{{x}} \left({x}+\mathrm{1}\right){dx}\:+\:\left(\mathrm{ye}^{\mathrm{y}} \:−\:{xe}^{{x}} \right)\mathrm{dy}=\mathrm{0} \\ $$

Question Number 41346 Answers: 0 Comments: 6

calculate ∫∫_([0,1]^2 ) cos(x^2 +y^2 )dxdy .

$${calculate}\:\int\int_{\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{2}} } \:{cos}\left({x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \right){dxdy}\:. \\ $$

Question Number 41343 Answers: 1 Comments: 1

calculate ∫∫_D (x^2 −y^2 )dxdy with D = [−1,1]^2

$${calculate}\:\:\:\:\int\int_{{D}} \:\left({x}^{\mathrm{2}} −{y}^{\mathrm{2}} \right){dxdy}\:\:{with} \\ $$$${D}\:=\:\left[−\mathrm{1},\mathrm{1}\right]^{\mathrm{2}} \\ $$

Question Number 41326 Answers: 0 Comments: 0

Evaluate ∫_0 ^(π/2) x^3 sec^5 x dx

$${Evaluate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {x}^{\mathrm{3}} {sec}^{\mathrm{5}} {x}\:{dx} \\ $$

Question Number 41302 Answers: 1 Comments: 1

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

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

Question Number 41301 Answers: 1 Comments: 4

let f(x)=∫_0 ^∞ e^(−ax) ln(1+e^(−bx) )dx with a>0 and b>0 1) calculate (∂f/∂a)(x) 2) calculate (∂f/∂b)(x) 3)find the value of ∫_0 ^∞ e^(−2x) ln(1+e^(−x) )dx and ∫_0 ^∞ e^(−x) ln(1+e^(−2x) )dx .

$${let}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{ax}} {ln}\left(\mathrm{1}+{e}^{−{bx}} \right){dx}\:{with}\:{a}>\mathrm{0}\:{and} \\ $$$${b}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\frac{\partial{f}}{\partial{a}}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\frac{\partial{f}}{\partial{b}}\left({x}\right) \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−\mathrm{2}{x}} {ln}\left(\mathrm{1}+{e}^{−{x}} \right){dx}\:{and} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{x}} {ln}\left(\mathrm{1}+{e}^{−\mathrm{2}{x}} \right){dx}\:. \\ $$

Question Number 41280 Answers: 1 Comments: 1

find f(x) = ∫_0 ^1 arctan(xt^2 )dt

$${find}\:\:{f}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{arctan}\left({xt}^{\mathrm{2}} \right){dt} \\ $$

Question Number 41279 Answers: 0 Comments: 1

let f(x)=∫_0 ^∞ arctan(xt^2 )dt . find a explicite form of f^′ (x)

$${let}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\:{arctan}\left({xt}^{\mathrm{2}} \right){dt}\:. \\ $$$${find}\:\:{a}\:{explicite}\:{form}\:{of}\:{f}^{'} \left({x}\right) \\ $$

Question Number 41273 Answers: 0 Comments: 2

find f(x)=∫_0 ^(+∞) arctan(xt^2 )dt with x fromR .

$${find}\:\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{+\infty} \:{arctan}\left({xt}^{\mathrm{2}} \right){dt}\:\:{with}\:{x}\:{fromR}\:. \\ $$

Question Number 41248 Answers: 2 Comments: 4