Question and Answers Forum

IntegrationQuestion and Answers: Page 222

Question Number 67011 Answers: 0 Comments: 1

calculate U_n =∫_1 ^(+∞) ((arctan(n[x]))/x^2 )dx

$${calculate}\:{U}_{{n}} =\int_{\mathrm{1}} ^{+\infty} \:\:\frac{{arctan}\left({n}\left[{x}\right]\right)}{{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 67008 Answers: 0 Comments: 2

let f(x) =∫_0 ^∞ (dt/((x^2 +t^2 )^2 )) with x>0 1) find a explicit form of (x) 2)find also g(x) =∫_0 ^∞ (dt/((x^2 +t^2 )^3 )) 3)find the values of integrals ∫_0 ^∞ (dt/((t^2 +3)^2 )) and ∫_0 ^∞ (dt/((t^2 +3)^3 )) 4) calculate U_θ =∫_0 ^∞ (dt/((t^2 +cos^2 θ)^2 )) with 0<θ<(π/2) 5) find f^((n)) (x) and f^((n)) (0) 6) developp f at integr serie

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dt}}{\left({x}^{\mathrm{2}} \:+{t}^{\mathrm{2}} \right)^{\mathrm{2}} }\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:\left({x}\right) \\ $$$$\left.\mathrm{2}\right){find}\:{also}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dt}}{\left({x}^{\mathrm{2}} \:+{t}^{\mathrm{2}} \right)^{\mathrm{3}} } \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{values}\:{of}\:{integrals}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dt}}{\left({t}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{2}} }\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dt}}{\left({t}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{3}} } \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:{U}_{\theta} =\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dt}}{\left({t}^{\mathrm{2}} \:+{cos}^{\mathrm{2}} \theta\right)^{\mathrm{2}} }\:\:\:{with}\:\mathrm{0}<\theta<\frac{\pi}{\mathrm{2}} \\ $$$$\left.\mathrm{5}\right)\:{find}\:{f}^{\left({n}\right)} \left({x}\right)\:{and}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{6}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$

Question Number 67006 Answers: 0 Comments: 1

calculae A_n =∫_0 ^∞ (dx/((x^2 +1)^n )) with n integr natural and n>0

$${calculae}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{{n}} }\:\:\:{with}\:{n}\:{integr}\:{natural}\:{and}\:{n}>\mathrm{0} \\ $$

Question Number 67005 Answers: 0 Comments: 1

find ∫ ((x−2(√(x^2 −1)))/(x+2(√(x^2 −1))))dx

$${find}\:\int\:\:\:\frac{{x}−\mathrm{2}\sqrt{{x}^{\mathrm{2}} −\mathrm{1}}}{{x}+\mathrm{2}\sqrt{{x}^{\mathrm{2}} −\mathrm{1}}}{dx} \\ $$

Question Number 66959 Answers: 0 Comments: 0

Question Number 66938 Answers: 2 Comments: 7

Question Number 66814 Answers: 0 Comments: 0

Let consider an integer serie {a_n x^n } given by a_n = H_n =Σ_(k=1) ^n (1/k) 1) Find out the largest domain D of convergence of that integer serie 2) ∀ x∈D , explicit the sum S(x) of the {a_n x^n } 3) Calculate ∫_(−1) ^1 S(1−x)S(x) dx .

$${Let}\:{consider}\:{an}\:{integer}\:{serie}\:\left\{{a}_{{n}} {x}^{{n}} \right\}\:{given}\:{by}\:\:{a}_{{n}} \:=\:{H}_{{n}} =\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{k}}\: \\ $$$$\left.\mathrm{1}\right)\:{Find}\:{out}\:{the}\:{largest}\:{domain}\:{D}\:{of}\:{convergence}\:{of}\:{that}\:{integer}\:{serie} \\ $$$$\left.\mathrm{2}\right)\:\forall\:{x}\in{D}\:\:,\:{explicit}\:{the}\:{sum}\:{S}\left({x}\right)\:{of}\:{the}\:\left\{{a}_{{n}} {x}^{{n}} \right\}\: \\ $$$$\left.\mathrm{3}\right)\:{Calculate}\:\:\int_{−\mathrm{1}} ^{\mathrm{1}} \:{S}\left(\mathrm{1}−{x}\right){S}\left({x}\right)\:{dx}\:. \\ $$$$ \\ $$$$\:\:\:\: \\ $$

Question Number 66801 Answers: 0 Comments: 3

let f(x) =∫_0 ^2 (√(x+t^2 ))dt with x≥0 1) calculate f(x) 2)calculate g(x) =∫_0 ^2 (dt/(√(x+t^2 ))) 3)find the value[of ∫_0 ^2 (√(4+t^2 ))dt and ∫_0 ^2 (dt/(√(3+t^2 ))) 4) give g^′ (x) at form of integral.

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{2}} \sqrt{{x}+{t}^{\mathrm{2}} }{dt}\:\:\:{with}\:{x}\geqslant\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){calculate}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{2}} \:\frac{{dt}}{\sqrt{{x}+{t}^{\mathrm{2}} }} \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\left[{of}\:\int_{\mathrm{0}} ^{\mathrm{2}} \sqrt{\mathrm{4}+{t}^{\mathrm{2}} }{dt}\:{and}\:\int_{\mathrm{0}} ^{\mathrm{2}} \frac{{dt}}{\sqrt{\mathrm{3}+{t}^{\mathrm{2}} }}\right. \\ $$$$\left.\mathrm{4}\right)\:{give}\:{g}^{'} \left({x}\right)\:{at}\:{form}\:{of}\:{integral}. \\ $$

Question Number 66795 Answers: 0 Comments: 3

let f(x) =e^(−x) ln(1+x^2 ) 1) calculate f^((n)) (0) 2) developp f at integr serie

$${let}\:{f}\left({x}\right)\:={e}^{−{x}} {ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right) \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$

Question Number 66794 Answers: 0 Comments: 1

calculate ∫_0 ^∞ ((cos(2arctan(2x)))/(9+x^2 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left(\mathrm{2}{arctan}\left(\mathrm{2}{x}\right)\right)}{\mathrm{9}+{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 66793 Answers: 0 Comments: 0

calculate ∫_0 ^1 cos(3arctanx)dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{cos}\left(\mathrm{3}{arctanx}\right){dx} \\ $$

Question Number 66792 Answers: 0 Comments: 1

calculate ∫_0 ^1 cos(2 arctan(x))dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{cos}\left(\mathrm{2}\:{arctan}\left({x}\right)\right){dx} \\ $$

Question Number 66790 Answers: 0 Comments: 0

find ∫_0 ^∞ (x/(sh(x)))dx

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}}{{sh}\left({x}\right)}{dx} \\ $$

Question Number 66787 Answers: 0 Comments: 0

find the value of ∫_0 ^∞ (x^2 /(ch(x)))dx

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{x}^{\mathrm{2}} }{{ch}\left({x}\right)}{dx} \\ $$

Question Number 66786 Answers: 0 Comments: 1

find the value of ∫_0 ^∞ (x/(ch(x)))dx

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{x}}{{ch}\left({x}\right)}{dx} \\ $$

Question Number 66740 Answers: 0 Comments: 1

∫_( 0) ^( 1) (√(1−x+x^2 −x^3 )) dx=?

$$\underset{\:\:\mathrm{0}} {\overset{\:\:\:\:\:\:\:\mathrm{1}} {\int}}\sqrt{\mathrm{1}−\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\boldsymbol{\mathrm{x}}^{\mathrm{3}} \:}\:\:\boldsymbol{\mathrm{dx}}=? \\ $$

Question Number 66739 Answers: 1 Comments: 0

find∫(√(dx ))

$${find}\int\sqrt{{dx}\:} \\ $$

Question Number 66731 Answers: 0 Comments: 1

y=x^2 −3x y=2x find area

$${y}={x}^{\mathrm{2}} −\mathrm{3}{x}\:\:\:\:\:{y}=\mathrm{2}{x}\:{find}\:{area} \\ $$

Question Number 66728 Answers: 2 Comments: 2

∫_1 ^∞ (1/(x(√(x^2 +1))))=?

$$\int_{\mathrm{1}} ^{\infty} \frac{\mathrm{1}}{{x}\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}}=? \\ $$

Question Number 66695 Answers: 1 Comments: 3

calculate ∫_0 ^∞ ((cos(arctanx))/(4+x^2 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({arctanx}\right)}{\mathrm{4}+{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 66694 Answers: 0 Comments: 3

let f(a) =∫_(−∞) ^(+∞) (dx/((x^4 +x^2 +a))) with a∈](1/4),+∞[ 1) calculate f(a) 2)find also g(a) =∫_(−∞) ^(+∞) (dx/((x^4 +x^2 +a)^2 )) 3) find the value of integrals ∫_0 ^∞ (dx/((x^4 +x^2 +3))) and ∫_0 ^∞ (dx/((x^4 +x^2 +1)^2 )) 4) developp f at integrserie.

$$\left.{let}\:{f}\left({a}\right)\:=\int_{−\infty} ^{+\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{4}} +{x}^{\mathrm{2}} \:+{a}\right)}\:{with}\:{a}\in\right]\frac{\mathrm{1}}{\mathrm{4}},+\infty\left[\right. \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right){find}\:{also}\:{g}\left({a}\right)\:=\int_{−\infty} ^{+\infty} \:\frac{{dx}}{\left({x}^{\mathrm{4}} \:+{x}^{\mathrm{2}} +{a}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:{integrals}\:\:\int_{\mathrm{0}} ^{\infty} \:\frac{{dx}}{\left({x}^{\mathrm{4}} \:+{x}^{\mathrm{2}} \:+\mathrm{3}\right)}\:{and} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{4}} \:+{x}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{4}\right)\:{developp}\:{f}\:{at}\:{integrserie}. \\ $$

Question Number 66693 Answers: 1 Comments: 1

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

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

Question Number 66627 Answers: 1 Comments: 4

Question Number 66589 Answers: 2 Comments: 5

∫((sinx)/(1+sinx+sin2x))dx

$$\int\frac{{sinx}}{\mathrm{1}+{sinx}+{sin}\mathrm{2}{x}}{dx} \\ $$

Question Number 66564 Answers: 0 Comments: 0

Question Number 66561 Answers: 1 Comments: 2

evaluate ∫_0 ^2 ∣ x+ 2∣ dx.

$${evaluate}\: \\ $$$$\:\:\int_{\mathrm{0}} ^{\mathrm{2}} \mid\:{x}+\:\mathrm{2}\mid\:{dx}. \\ $$

Pg 217 Pg 218 Pg 219 Pg 220 Pg 221 Pg 222 Pg 223 Pg 224 Pg 225 Pg 226

Contact: info@tinkutara.com