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Question Number 57750 Answers: 1 Comments: 0

find ∫ x^2 (√(25−x^2 ))dx

$${find}\:\int\:{x}^{\mathrm{2}} \sqrt{\mathrm{25}−{x}^{\mathrm{2}} }{dx}\: \\ $$

Question Number 57749 Answers: 1 Comments: 3

find ∫ (dx/(x^2 (√(9+x^2 ))))

$${find}\:\int\:\:\frac{{dx}}{{x}^{\mathrm{2}} \sqrt{\mathrm{9}+{x}^{\mathrm{2}} }} \\ $$

Question Number 57748 Answers: 2 Comments: 2

find ∫ x^2 (√(4+x^2 ))dx

$${find}\:\int\:{x}^{\mathrm{2}} \sqrt{\mathrm{4}+{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 57746 Answers: 0 Comments: 4

let f(x)=∫_(−∞) ^(+∞) (dt/((t^2 −2xt +1)^2 )) with ∣x∣<1 (x real) 1) determine a explicit form for f(x) 2) find also g(x) =∫_(−∞) ^(+∞) ((tdt)/((t^2 −2xt +1)^3 )) 3) calculate ∫_(−∞) ^(+∞) (dt/((t^2 −(√2)t +1)^2 )) and ∫_(−∞) ^(+∞) ((tdt)/((t^2 −(√2)t +1)^3 )) 4) calculate A(θ) =∫_(−∞) ^(+∞) (dt/((t^2 −2cosθ t+1)^2 )) and B(θ) =∫_(−∞) ^(+∞) ((tdt)/((t^2 −2cosθ t +1)^3 )) with 0<θ <(π/2) .

$${let}\:{f}\left({x}\right)=\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{dt}}{\left({t}^{\mathrm{2}} −\mathrm{2}{xt}\:+\mathrm{1}\right)^{\mathrm{2}} }\:\:{with}\:\mid{x}\mid<\mathrm{1}\:\:\:\left({x}\:{real}\right) \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:\:{for}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{also}\:{g}\left({x}\right)\:=\int_{−\infty} ^{+\infty} \:\:\:\frac{{tdt}}{\left({t}^{\mathrm{2}} −\mathrm{2}{xt}\:+\mathrm{1}\right)^{\mathrm{3}} } \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{dt}}{\left({t}^{\mathrm{2}} −\sqrt{\mathrm{2}}{t}\:+\mathrm{1}\right)^{\mathrm{2}} }\:\:\:{and}\:\int_{−\infty} ^{+\infty} \:\:\frac{{tdt}}{\left({t}^{\mathrm{2}} −\sqrt{\mathrm{2}}{t}\:+\mathrm{1}\right)^{\mathrm{3}} } \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:{A}\left(\theta\right)\:=\int_{−\infty} ^{+\infty} \:\:\:\frac{{dt}}{\left({t}^{\mathrm{2}} \:−\mathrm{2}{cos}\theta\:{t}+\mathrm{1}\right)^{\mathrm{2}} }\:\:\:{and}\: \\ $$$${B}\left(\theta\right)\:=\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{tdt}}{\left({t}^{\mathrm{2}} \:−\mathrm{2}{cos}\theta\:{t}\:+\mathrm{1}\right)^{\mathrm{3}} }\:\:\:\:{with}\:\mathrm{0}<\theta\:<\frac{\pi}{\mathrm{2}}\:\:\:\:\:. \\ $$

Question Number 57668 Answers: 0 Comments: 3

let V_n = ∫_0 ^(1+(1/n)) ((x+1)/(√(2x^2 +3))) dx 1) calculate lim_(n→+∞) V_n 2) find nature of the serie Σ V_n

$${let}\:{V}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}+\frac{\mathrm{1}}{{n}}} \:\:\:\:\frac{{x}+\mathrm{1}}{\sqrt{\mathrm{2}{x}^{\mathrm{2}} \:+\mathrm{3}}}\:{dx}\:\: \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{lim}_{{n}\rightarrow+\infty} \:{V}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{nature}\:{of}\:{the}\:{serie}\:\Sigma\:{V}_{{n}} \\ $$

Question Number 57667 Answers: 0 Comments: 3

calculate U_n =∫_(π/n) ^((2π)/n) (dx/(2 +sinx)) 1) calculate U_n and lim_(n→+∞) nU_n 2) find nature of Σ U_n

$${calculate}\:{U}_{{n}} =\int_{\frac{\pi}{{n}}} ^{\frac{\mathrm{2}\pi}{{n}}} \:\:\:\:\:\frac{{dx}}{\mathrm{2}\:+{sinx}} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{U}_{{n}} \:\:\:\:\:\:\:{and}\:{lim}_{{n}\rightarrow+\infty} \:\:{nU}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{nature}\:{of}\:\Sigma\:{U}_{{n}} \\ $$

Question Number 57666 Answers: 0 Comments: 3

1) calculate f(θ) =∫_0 ^1 (√(t^2 +2sinθt +1))dt with 0≤θ≤(π/2) 2) calculate g(t) =∫_0 ^1 (√(t^2 +2(sinθ)t +1))dθ 3) find also h(θ) =∫_0 ^1 (t/(√(t^2 +2(sinθ)t +1)))dt

$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left(\theta\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{{t}^{\mathrm{2}} \:+\mathrm{2}{sin}\theta{t}\:+\mathrm{1}}{dt}\:\:\:\:{with}\:\mathrm{0}\leqslant\theta\leqslant\frac{\pi}{\mathrm{2}} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{g}\left({t}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{{t}^{\mathrm{2}} \:+\mathrm{2}\left({sin}\theta\right){t}\:+\mathrm{1}}{d}\theta \\ $$$$\left.\mathrm{3}\right)\:{find}\:{also}\:{h}\left(\theta\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{t}}{\sqrt{{t}^{\mathrm{2}} \:+\mathrm{2}\left({sin}\theta\right){t}\:+\mathrm{1}}}{dt} \\ $$

Question Number 57665 Answers: 0 Comments: 4

let f(a) =∫_(π/4) ^(π/3) (√(a+tan^2 x))dx with a>0 1) find a explicit form of f(a) 2) find also g(a) =∫_(π/4) ^(π/3) (dx/(√(a+tan^2 x))) 3) find the values of ∫_(π/4) ^(π/3) (√(2+tan^2 x))dx and ∫_(π/4) ^(π/3) (dx/(√(3+tan^2 x)))

$${let}\:{f}\left({a}\right)\:=\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \sqrt{{a}+{tan}^{\mathrm{2}} {x}}{dx}\:\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:\:{find}\:{also}\:{g}\left({a}\right)\:=\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \:\:\frac{{dx}}{\sqrt{{a}+{tan}^{\mathrm{2}} {x}}} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{values}\:{of}\:\:\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \:\sqrt{\mathrm{2}+{tan}^{\mathrm{2}} {x}}{dx}\:\:{and}\:\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \:\:\:\frac{{dx}}{\sqrt{\mathrm{3}+{tan}^{\mathrm{2}} {x}}} \\ $$

Question Number 57653 Answers: 0 Comments: 5

is it possible to find the exact value of I? I=∫_0 ^π sin (sin x) dx

$$\mathrm{is}\:\mathrm{it}\:\mathrm{possible}\:\mathrm{to}\:\mathrm{find}\:\mathrm{the}\:\mathrm{exact}\:\mathrm{value}\:\mathrm{of}\:{I}? \\ $$$${I}=\underset{\mathrm{0}} {\overset{\pi} {\int}}\mathrm{sin}\:\left(\mathrm{sin}\:{x}\right)\:{dx} \\ $$

Question Number 57490 Answers: 1 Comments: 2

1)findF(a)= ∫_0 ^∞ ((cos(ln(2+x^2 )))/(a^2 +x^2 ))dx witha>0 2) find the value of ∫_0 ^∞ ((cos(ln(2+x^2 )))/(4+x^2 ))dx.

$$\left.\mathrm{1}\right){findF}\left({a}\right)=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{cos}\left({ln}\left(\mathrm{2}+{x}^{\mathrm{2}} \right)\right)}{{a}^{\mathrm{2}} \:+{x}^{\mathrm{2}} }{dx}\:\:{witha}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{cos}\left({ln}\left(\mathrm{2}+{x}^{\mathrm{2}} \right)\right)}{\mathrm{4}+{x}^{\mathrm{2}} }{dx}. \\ $$

Question Number 57487 Answers: 0 Comments: 1

calculate lim_(x→1) ∫_x ^x^2 ((arctan(t))/(sint))dt .

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{1}} \:\:\:\int_{{x}} ^{{x}^{\mathrm{2}} } \:\:\:\:\frac{{arctan}\left({t}\right)}{{sint}}{dt}\:. \\ $$

Question Number 57423 Answers: 0 Comments: 0

let A_n =∫_0 ^∞ (dt/((e^t +e^(−t) )^n )) calculate A_n interms of n

$${let}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dt}}{\left({e}^{{t}} \:+\overset{−{t}} {{e}}\right)^{{n}} } \\ $$$${calculate}\:{A}_{{n}} \:{interms}\:{of}\:{n} \\ $$

Question Number 57421 Answers: 1 Comments: 0

calculate ∫_(−1) ^1 (((x^4 +x^2 +1)^2 +e^x )/(e^x +1))dx

$${calculate}\:\:\int_{−\mathrm{1}} ^{\mathrm{1}} \:\:\:\frac{\left({x}^{\mathrm{4}} \:+{x}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{2}} \:+{e}^{{x}} }{{e}^{{x}} \:+\mathrm{1}}{dx} \\ $$

Question Number 57420 Answers: 0 Comments: 1

let J(x)=∫_0 ^x (t^2 /((√(t+1)) +(√(t+4))))dt find a explicit form of J(x)

$${let}\:{J}\left({x}\right)=\int_{\mathrm{0}} ^{{x}} \:\:\:\:\frac{{t}^{\mathrm{2}} }{\sqrt{{t}+\mathrm{1}}\:+\sqrt{{t}+\mathrm{4}}}{dt} \\ $$$${find}\:{a}\:{explicit}\:{form}\:{of}\:{J}\left({x}\right) \\ $$

Question Number 57419 Answers: 0 Comments: 1

find ∫_0 ^1 (x+1) ln(x+(√(1+x^2 )))dx

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \left({x}+\mathrm{1}\right)\:{ln}\left({x}+\sqrt{\left.\mathrm{1}+{x}^{\mathrm{2}} \right)}{dx}\right. \\ $$

Question Number 57418 Answers: 0 Comments: 1

calculate ∫_(−1) ^4 ((∣x−1∣+∣x−2∣)/(∣x^2 −9∣ +x^2 +16))dx

$${calculate}\:\int_{−\mathrm{1}} ^{\mathrm{4}} \:\frac{\mid{x}−\mathrm{1}\mid+\mid{x}−\mathrm{2}\mid}{\mid{x}^{\mathrm{2}} −\mathrm{9}\mid\:+{x}^{\mathrm{2}} \:+\mathrm{16}}{dx} \\ $$

Question Number 57417 Answers: 0 Comments: 2

let F(x) =∫_0 ^x ((1+sint)/(2+cost))dt 1) find a explicite form of f(x) 2) calculate ∫_0 ^π ((1+sint)/(2+cost))dt

$${let}\:{F}\left({x}\right)\:=\int_{\mathrm{0}} ^{{x}} \:\:\frac{\mathrm{1}+{sint}}{\mathrm{2}+{cost}}{dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicite}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{\mathrm{1}+{sint}}{\mathrm{2}+{cost}}{dt} \\ $$

Question Number 57388 Answers: 0 Comments: 0

Given f(x) = f(x + 2016), ∀x ∈ R If ∫_0 ^3 f(x) = 30, then ∫_3 ^5 f(x + 2016) = ...

$$\mathrm{Given}\:{f}\left({x}\right)\:=\:{f}\left({x}\:+\:\mathrm{2016}\right),\:\:\forall{x}\:\in\:\mathbb{R} \\ $$$$\mathrm{If}\:\underset{\mathrm{0}} {\overset{\mathrm{3}} {\int}}\:{f}\left({x}\right)\:=\:\mathrm{30},\:\mathrm{then}\:\underset{\mathrm{3}} {\overset{\mathrm{5}} {\int}}\:{f}\left({x}\:+\:\mathrm{2016}\right)\:=\:... \\ $$

Question Number 57385 Answers: 1 Comments: 1

Question Number 57325 Answers: 0 Comments: 1

calculate ∫_0 ^(π/2) ((ln(1+sinx))/(sinx))dx

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{{ln}\left(\mathrm{1}+{sinx}\right)}{{sinx}}{dx} \\ $$

Question Number 57324 Answers: 0 Comments: 0

we want to find the vslue of I =∫_0 ^1 ((ln(1+x))/(1+x^2 )) dx let A=∫∫_W (x/((1+x^2 )(1+xy)))dxdy with W=[0,1]^2 calculate A by two method and conclude the value of I .

$${we}\:{want}\:{to}\:{find}\:{the}\:{vslue}\:{of} \\ $$$${I}\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}+{x}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx}\:{let} \\ $$$${A}=\int\int_{{W}} \frac{{x}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{xy}\right)}{dxdy} \\ $$$${with}\:{W}=\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{2}} \\ $$$${calculate}\:{A}\:{by}\:{two}\:{method}\:{and} \\ $$$${conclude}\:{the}\:{value}\:{of}\:{I}\:. \\ $$

Question Number 57323 Answers: 0 Comments: 1

calculate ∫∫_D ((x+y)/(3+(√(x^2 +y^2 ))))dxdy with D={(x,y)∈R^2 /x^2 +y^2 ≤2 and x≥0 ,y≥0}

$${calculate}\:\int\int_{{D}} \:\:\frac{{x}+{y}}{\mathrm{3}+\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }}{dxdy} \\ $$$${with}\:{D}=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \leqslant\mathrm{2}\right. \\ $$$$\left.{and}\:{x}\geqslant\mathrm{0}\:,{y}\geqslant\mathrm{0}\right\} \\ $$

Question Number 57321 Answers: 1 Comments: 1

calculate ∫∫_D (x−y)(√(x^2 +y^2 ))dxdy with D ={ (x,y)∈R^2 /x^2 +y^2 ≤2 and x≥0}

$${calculate}\:\int\int_{{D}} \left({x}−{y}\right)\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }{dxdy} \\ $$$${with}\:{D}\:=\left\{\:\left({x},{y}\right)\in{R}^{\mathrm{2}} /{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:\leqslant\mathrm{2}\:{and}\:{x}\geqslant\mathrm{0}\right\} \\ $$

Question Number 57320 Answers: 1 Comments: 1

calculate ∫∫_D xy e^(−x^2 −y^2 ) dxdy with D={(x,y)∈R^2 / 0≤x≤2 and 1≤y≤3}

$${calculate}\:\int\int_{{D}} {xy}\:{e}^{−{x}^{\mathrm{2}} −{y}^{\mathrm{2}} } \:{dxdy} \\ $$$${with}\:{D}=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /\:\mathrm{0}\leqslant{x}\leqslant\mathrm{2}\:{and}\right. \\ $$$$\left.\mathrm{1}\leqslant{y}\leqslant\mathrm{3}\right\} \\ $$

Question Number 57319 Answers: 1 Comments: 1

calculate ∫∫_D e^(x−y) dxdy with D={(x,y)∈R^2 /∣x∣<1 and 0≤y≤1}

$${calculate}\:\int\int_{{D}} \:{e}^{{x}−{y}} \:{dxdy} \\ $$$${with}\:{D}=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} \:/\mid{x}\mid<\mathrm{1}\:{and}\:\mathrm{0}\leqslant{y}\leqslant\mathrm{1}\right\} \\ $$

Question Number 57228 Answers: 0 Comments: 1

find f(x) =∫_1 ^2 ((ln(1+xt))/t^2 ) dt with x>0

$${find}\:{f}\left({x}\right)\:=\int_{\mathrm{1}} ^{\mathrm{2}} \:\frac{{ln}\left(\mathrm{1}+{xt}\right)}{{t}^{\mathrm{2}} }\:{dt}\:\:{with}\:{x}>\mathrm{0} \\ $$

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