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IntegrationQuestion and Answers: Page 247

Question Number 53271    Answers: 0   Comments: 2

1)calculate∫_0 ^∞ e^(−xt^2 ) dt with x>0 2) find the value of ∫_0 ^∞ ((e^(−t^2 ) −e^(−2t^2 ) )/t^2 ) dt by using fubinni theorem .

$$\left.\mathrm{1}\right){calculate}\int_{\mathrm{0}} ^{\infty} \:\:\:{e}^{−{xt}^{\mathrm{2}} } {dt}\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{t}^{\mathrm{2}} } \:−{e}^{−\mathrm{2}{t}^{\mathrm{2}} } }{{t}^{\mathrm{2}} }\:{dt}\:\:{by}\:{using} \\ $$$${fubinni}\:{theorem}\:. \\ $$

Question Number 53270    Answers: 1   Comments: 1

1)calculate ∫_0 ^∞ e^(−at) dt with a>0 2)by using fubinni theorem find the value of ∫_0 ^∞ ((e^(−t) −e^(−xt) )/t)dt with x>0 .

$$\left.\mathrm{1}\right){calculate}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{at}} {dt}\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right){by}\:{using}\:{fubinni}\:{theorem}\:{find}\:{the}\:{value}\:{of} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{t}} \:−{e}^{−{xt}} }{{t}}{dt}\:\:\:{with}\:{x}>\mathrm{0}\:. \\ $$

Question Number 53261    Answers: 0   Comments: 0

1)find f(x)=∫_0 ^1 e^(−2t) ln(1−xt)dt with ∣x∣<1 2) calculate ∫_0 ^1 e^(−2t) ln(1−((t(√2))/2))dt.

$$\left.\mathrm{1}\right){find}\:\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:{e}^{−\mathrm{2}{t}} {ln}\left(\mathrm{1}−{xt}\right){dt}\:{with}\:\mid{x}\mid<\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{e}^{−\mathrm{2}{t}} {ln}\left(\mathrm{1}−\frac{{t}\sqrt{\mathrm{2}}}{\mathrm{2}}\right){dt}. \\ $$

Question Number 53259    Answers: 1   Comments: 0

Question Number 53228    Answers: 0   Comments: 3

1) find f(a) =∫_0 ^1 (dx/((ax+1)(√(x^2 −x+1)))) with a>0 2) calculate f^′ (a) 3)find the value of ∫_0 ^1 ((xdx)/((ax+1)^2 (√(x^2 −x+1)))) 4) calculate ∫_0 ^1 (dx/((2x+1)(√(x^2 −x+1)))) and ∫_0 ^1 ((xdx)/((2x+1)^2 (√(x^2 −x+1))))

$$\left.\mathrm{1}\right)\:{find}\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dx}}{\left({ax}+\mathrm{1}\right)\sqrt{{x}^{\mathrm{2}} −{x}+\mathrm{1}}}\:\:\:{with}\:\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{f}^{'} \left({a}\right) \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{xdx}}{\left({ax}+\mathrm{1}\right)^{\mathrm{2}} \sqrt{{x}^{\mathrm{2}} −{x}+\mathrm{1}}} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dx}}{\left(\mathrm{2}{x}+\mathrm{1}\right)\sqrt{{x}^{\mathrm{2}} −{x}+\mathrm{1}}}\:{and}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{xdx}}{\left(\mathrm{2}{x}+\mathrm{1}\right)^{\mathrm{2}} \sqrt{{x}^{\mathrm{2}} −{x}+\mathrm{1}}} \\ $$

Question Number 53212    Answers: 2   Comments: 21

Let f(x) = ((2x)/(x^2 + 4)) (a) Find ∫_(−b) ^b f(x) dx, for b > 0 (b) Determine ∫_(−∞) ^∞ f(x) dx is convergent or not

$$\mathrm{Let}\:{f}\left({x}\right)\:=\:\frac{\mathrm{2}{x}}{{x}^{\mathrm{2}} \:+\:\mathrm{4}} \\ $$$$ \\ $$$$\left({a}\right)\:\mathrm{Find}\:\underset{−{b}} {\overset{{b}} {\int}}\:{f}\left({x}\right)\:{dx},\:\mathrm{for}\:{b}\:>\:\mathrm{0} \\ $$$$\left({b}\right)\:\mathrm{Determine}\:\underset{−\infty} {\overset{\infty} {\int}}\:{f}\left({x}\right)\:{dx}\:\mathrm{is}\:\mathrm{convergent}\:\mathrm{or}\:\mathrm{not} \\ $$

Question Number 53119    Answers: 6   Comments: 3

Evaluate : 1) ∫(√((2−x)/(4+x))) dx 2) ∫ (√((x−2)/(x−4))) dx 3) ∫ (√((x−2)(x−4))) dx 4) ∫ (dx/(2sinx+3secx)) .

$${Evaluate}\:: \\ $$$$\left.\mathrm{1}\right)\:\int\sqrt{\frac{\mathrm{2}−{x}}{\mathrm{4}+{x}}}\:{dx} \\ $$$$\left.\mathrm{2}\right)\:\int\:\sqrt{\frac{{x}−\mathrm{2}}{{x}−\mathrm{4}}}\:{dx} \\ $$$$\left.\mathrm{3}\right)\:\int\:\sqrt{\left({x}−\mathrm{2}\right)\left({x}−\mathrm{4}\right)}\:{dx} \\ $$$$\left.\mathrm{4}\right)\:\int\:\frac{{dx}}{\mathrm{2sin}\boldsymbol{{x}}+\mathrm{3sec}\boldsymbol{{x}}}\:. \\ $$

Question Number 53118    Answers: 1   Comments: 0

If a<∫_0 ^(2π) (1/(10+3 cos x)) dx<b, then the ordered pair (a, b) is

$$\mathrm{If}\:{a}<\int_{\mathrm{0}} ^{\mathrm{2}\pi} \frac{\mathrm{1}}{\mathrm{10}+\mathrm{3}\:\mathrm{cos}\:{x}}\:{dx}<{b},\:\mathrm{then}\:\mathrm{the} \\ $$$$\mathrm{ordered}\:\mathrm{pair}\:\left({a},\:{b}\right)\:\mathrm{is} \\ $$

Question Number 53114    Answers: 0   Comments: 1

let A_n =∫_0 ^∞ ((x sin(nx))/((x^2 +n^2 )^2 ))dx with n integr natural not 0 1) find the value of A_n 2) study the convergence of Σ A_n

$${let}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{x}\:{sin}\left({nx}\right)}{\left({x}^{\mathrm{2}} \:+{n}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx}\:\:{with}\:{n}\:{integr}\:{natural}\:{not}\:\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{value}\:{of}\:\:{A}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{study}\:{the}\:{convergence}\:{of}\:\Sigma\:{A}_{{n}} \\ $$

Question Number 53113    Answers: 0   Comments: 1

let I =∫_(−∞) ^(+∞) ((t+1)/((t^2 −t+1)^2 ))dt find value of I .

$${let}\:{I}\:=\int_{−\infty} ^{+\infty} \:\:\:\frac{{t}+\mathrm{1}}{\left({t}^{\mathrm{2}} −{t}+\mathrm{1}\right)^{\mathrm{2}} }{dt} \\ $$$${find}\:{value}\:{of}\:{I}\:. \\ $$

Question Number 53112    Answers: 1   Comments: 0

calculate ∫_0 ^π ((1+2sinx)/(3 +2cosx))dx let A =∫_0 ^π ((1+2sinx)/(3 +2cosx))dx changement tan((x/2))=t give A =∫_0 ^∞ ((1+((4t)/(1+t^2 )))/(3+2((1−t^2 )/(1+t^2 )))) ((2dt)/(1+t^2 )) =2 ∫_0 ^∞ ((1+t^2 +4t)/((1+t^2 )^2 (((3+3t^2 +2−2t^2 )/(1+t^2 )))))dt =2 ∫_0 ^∞ ((t^2 +4t +1)/((1+t^2 )(5+t^2 )))dt let decompose F(t)=((t^2 +4t+1)/((t^2 +1)(t^2 +5))) F(t)=((at +b)/(t^2 +1)) +((ct +d)/(t^2 +5)) ⇒(at+b)(t^2 +5)+(ct+d)(t^2 +1) =t^2 +4t +1 ⇒ at^3 +5at +bt^2 +5b +ct^3 +ct +dt^2 +d =t^2 +4t +1 ⇒ (a+c)t^3 +(b+d)t^2 +(5a+c)t +5b +d =t^2 +4t +1 ⇒a+c=0 and b+d=1 and 5a+c =4 and 5b+d =1 ⇒c=−a ⇒a=1 ⇒c=−1 we have d=1−b ⇒5b +1−b =1 ⇒b=0 ⇒d=1 ⇒ F(t)=(t/(t^2 +1)) +((−t +1)/(t^2 +5)) ⇒ A =2 ∫_0 ^∞ F(t)dt =∫_0 ^∞ ((2t)/(t^2 +1))dt +∫_0 ^∞ ((−2t +2)/(t^2 +5))dt =[ln(((t^2 +1)/(t^2 +5)))]_0 ^(+∞) +2 ∫_0 ^∞ (dt/(t^2 +5)) =ln(5) + 2 ∫_0 ^∞ (dt/(t^2 +5)) but ∫_0 ^∞ (dt/(t^2 +5))dt =_(t =(√5)u ) ∫_0 ^∞ (((√5)du)/(5(1+u^2 ))) =(1/(√5)) [artanu]_0 ^(+∞) =(π/(2(√5))) ⇒ A =ln(5) +(π/(2(√5))) .

$${calculate}\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{\mathrm{1}+\mathrm{2}{sinx}}{\mathrm{3}\:+\mathrm{2}{cosx}}{dx} \\ $$$${let}\:{A}\:=\int_{\mathrm{0}} ^{\pi} \:\frac{\mathrm{1}+\mathrm{2}{sinx}}{\mathrm{3}\:+\mathrm{2}{cosx}}{dx}\:\:{changement}\:{tan}\left(\frac{{x}}{\mathrm{2}}\right)={t}\:{give} \\ $$$${A}\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{\mathrm{1}+\frac{\mathrm{4}{t}}{\mathrm{1}+{t}^{\mathrm{2}} }}{\mathrm{3}+\mathrm{2}\frac{\mathrm{1}−{t}^{\mathrm{2}} }{\mathrm{1}+{t}^{\mathrm{2}} }}\:\frac{\mathrm{2}{dt}}{\mathrm{1}+{t}^{\mathrm{2}} }\:=\mathrm{2}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{\mathrm{1}+{t}^{\mathrm{2}} \:+\mathrm{4}{t}}{\left(\mathrm{1}+{t}^{\mathrm{2}} \right)^{\mathrm{2}} \left(\frac{\mathrm{3}+\mathrm{3}{t}^{\mathrm{2}} +\mathrm{2}−\mathrm{2}{t}^{\mathrm{2}} }{\mathrm{1}+{t}^{\mathrm{2}} }\right)}{dt} \\ $$$$=\mathrm{2}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{t}^{\mathrm{2}} +\mathrm{4}{t}\:+\mathrm{1}}{\left(\mathrm{1}+{t}^{\mathrm{2}} \right)\left(\mathrm{5}+{t}^{\mathrm{2}} \right)}{dt}\:\:{let}\:{decompose}\:{F}\left({t}\right)=\frac{{t}^{\mathrm{2}} \:+\mathrm{4}{t}+\mathrm{1}}{\left({t}^{\mathrm{2}} +\mathrm{1}\right)\left({t}^{\mathrm{2}} \:+\mathrm{5}\right)} \\ $$$${F}\left({t}\right)=\frac{{at}\:+{b}}{{t}^{\mathrm{2}} \:+\mathrm{1}}\:+\frac{{ct}\:+{d}}{{t}^{\mathrm{2}} \:+\mathrm{5}}\:\Rightarrow\left({at}+{b}\right)\left({t}^{\mathrm{2}} \:+\mathrm{5}\right)+\left({ct}+{d}\right)\left({t}^{\mathrm{2}} \:+\mathrm{1}\right)\:={t}^{\mathrm{2}} \:+\mathrm{4}{t}\:+\mathrm{1}\:\Rightarrow \\ $$$${at}^{\mathrm{3}} \:+\mathrm{5}{at}\:+{bt}^{\mathrm{2}} \:+\mathrm{5}{b}\:+{ct}^{\mathrm{3}} \:+{ct}\:+{dt}^{\mathrm{2}} \:+{d}\:={t}^{\mathrm{2}} \:+\mathrm{4}{t}\:+\mathrm{1}\:\Rightarrow \\ $$$$\left({a}+{c}\right){t}^{\mathrm{3}} \:+\left({b}+{d}\right){t}^{\mathrm{2}} \:+\left(\mathrm{5}{a}+{c}\right){t}\:+\mathrm{5}{b}\:+{d}\:={t}^{\mathrm{2}} \:+\mathrm{4}{t}\:+\mathrm{1}\:\Rightarrow{a}+{c}=\mathrm{0}\:{and}\:{b}+{d}=\mathrm{1}\:{and} \\ $$$$\mathrm{5}{a}+{c}\:=\mathrm{4}\:{and}\:\mathrm{5}{b}+{d}\:=\mathrm{1}\:\Rightarrow{c}=−{a}\:\Rightarrow{a}=\mathrm{1}\:\Rightarrow{c}=−\mathrm{1}\: \\ $$$${we}\:{have}\:{d}=\mathrm{1}−{b}\:\Rightarrow\mathrm{5}{b}\:+\mathrm{1}−{b}\:=\mathrm{1}\:\Rightarrow{b}=\mathrm{0}\:\Rightarrow{d}=\mathrm{1}\:\Rightarrow \\ $$$${F}\left({t}\right)=\frac{{t}}{{t}^{\mathrm{2}} \:+\mathrm{1}}\:+\frac{−{t}\:+\mathrm{1}}{{t}^{\mathrm{2}} \:+\mathrm{5}}\:\:\Rightarrow\:{A}\:=\mathrm{2}\:\int_{\mathrm{0}} ^{\infty} \:{F}\left({t}\right){dt}\:=\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{2}{t}}{{t}^{\mathrm{2}} \:+\mathrm{1}}{dt}\:+\int_{\mathrm{0}} ^{\infty} \:\frac{−\mathrm{2}{t}\:+\mathrm{2}}{{t}^{\mathrm{2}} \:+\mathrm{5}}{dt} \\ $$$$=\left[{ln}\left(\frac{{t}^{\mathrm{2}} \:+\mathrm{1}}{{t}^{\mathrm{2}} \:+\mathrm{5}}\right)\right]_{\mathrm{0}} ^{+\infty} \:\:+\mathrm{2}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dt}}{{t}^{\mathrm{2}} \:+\mathrm{5}}\:={ln}\left(\mathrm{5}\right)\:+\:\mathrm{2}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dt}}{{t}^{\mathrm{2}} \:+\mathrm{5}}\:\:{but} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dt}}{{t}^{\mathrm{2}} \:+\mathrm{5}}{dt}\:=_{{t}\:=\sqrt{\mathrm{5}}{u}\:} \:\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\sqrt{\mathrm{5}}{du}}{\mathrm{5}\left(\mathrm{1}+{u}^{\mathrm{2}} \right)}\:=\frac{\mathrm{1}}{\sqrt{\mathrm{5}}}\:\left[{artanu}\right]_{\mathrm{0}} ^{+\infty} \:=\frac{\pi}{\mathrm{2}\sqrt{\mathrm{5}}}\:\Rightarrow \\ $$$${A}\:={ln}\left(\mathrm{5}\right)\:+\frac{\pi}{\mathrm{2}\sqrt{\mathrm{5}}}\:. \\ $$

Question Number 53081    Answers: 3   Comments: 0

Question Number 53080    Answers: 1   Comments: 1

calculate ∫_0 ^π ((cos^2 x)/(2+3sin(2x)))dx

$${calculate}\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{{cos}^{\mathrm{2}} {x}}{\mathrm{2}+\mathrm{3}{sin}\left(\mathrm{2}{x}\right)}{dx} \\ $$

Question Number 53078    Answers: 1   Comments: 1

∫_0 ^1 (1/((x^3 +1)^(3/2) )) dx=...

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\left({x}^{\mathrm{3}} +\mathrm{1}\right)^{\mathrm{3}/\mathrm{2}} }\:{dx}=... \\ $$

Question Number 53089    Answers: 0   Comments: 0

∫_( (π/2) ) ^( ∞) (dx/((5 + x^2 ) tanh^(−1) ((x/3))))

$$\int_{\:\frac{\pi}{\mathrm{2}}\:} ^{\:\infty} \:\:\:\frac{\mathrm{dx}}{\left(\mathrm{5}\:+\:\mathrm{x}^{\mathrm{2}} \right)\:\mathrm{tanh}^{−\mathrm{1}} \left(\frac{\mathrm{x}}{\mathrm{3}}\right)} \\ $$

Question Number 52999    Answers: 0   Comments: 6

∫_0 ^( ∞) ((x ln^2 (x))/(e^x − 1)) dx

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

Question Number 52988    Answers: 1   Comments: 0

∫ (x^2 /(√(1 + x^4 ))) dx

$$\int\:\frac{{x}^{\mathrm{2}} }{\sqrt{\mathrm{1}\:+\:{x}^{\mathrm{4}} }}\:{dx} \\ $$

Question Number 52944    Answers: 1   Comments: 0

∫_( 0) ^( 1) ((x^3 − 1)/((1 + x^2 ) ln x)) dx

$$\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\:\frac{\boldsymbol{\mathrm{x}}^{\mathrm{3}} \:−\:\mathrm{1}}{\left(\mathrm{1}\:+\:\boldsymbol{\mathrm{x}}^{\mathrm{2}} \right)\:\boldsymbol{\mathrm{ln}}\:\boldsymbol{\mathrm{x}}}\:\:\boldsymbol{\mathrm{dx}} \\ $$

Question Number 52898    Answers: 1   Comments: 0

∫arcsin x arccos x dx=?

$$\int\mathrm{arcsin}\:{x}\:\mathrm{arccos}\:{x}\:{dx}=? \\ $$

Question Number 52900    Answers: 3   Comments: 0

∫_0 ^(π/2) sin x (√(sin 2x)) dx=? ∫_(−(π/4)) ^(π/4) cos x (√(cos 2x)) dx=?

$$\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\mathrm{sin}\:{x}\:\sqrt{\mathrm{sin}\:\mathrm{2}{x}}\:{dx}=? \\ $$$$\underset{−\frac{\pi}{\mathrm{4}}} {\overset{\frac{\pi}{\mathrm{4}}} {\int}}\mathrm{cos}\:{x}\:\sqrt{\mathrm{cos}\:\mathrm{2}{x}}\:{dx}=? \\ $$

Question Number 52703    Answers: 0   Comments: 1

let f(t) =∫_0 ^∞ ((cos^2 (tx))/((x^2 +3)^2 )) dx with t ≥0 1) give a explicit form of f(t) 2) find the value of ∫_0 ^∞ ((xsin(2tx))/((x^2 +3)^2 )) dx 3) give the values of integrals ∫_0 ^∞ (dx/((x^2 +3)^2 )) and ∫_0 ^∞ ((cos^2 (πx))/((x^2 +3)^2 ))dx 4) give the values of integrals ∫_0 ^∞ ((xsin(πx))/((x^2 +3)^2 )) and ∫_0 ^∞ ((xsin(((πx)/2)))/((x^2 +3)^2 )) dx .

$${let}\:{f}\left({t}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}^{\mathrm{2}} \left({tx}\right)}{\left({x}^{\mathrm{2}} +\mathrm{3}\right)^{\mathrm{2}} }\:{dx}\:\:{with}\:{t}\:\geqslant\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{give}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({t}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{xsin}\left(\mathrm{2}{tx}\right)}{\left({x}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{2}} }\:{dx} \\ $$$$\left.\mathrm{3}\right)\:{give}\:{the}\:{values}\:{of}\:{integrals}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{2}} }\:{and}\:\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}^{\mathrm{2}} \left(\pi{x}\right)}{\left({x}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{4}\right)\:{give}\:{the}\:{values}\:{of}\:{integrals}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{xsin}\left(\pi{x}\right)}{\left({x}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{2}} }\:{and}\:\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{xsin}\left(\frac{\pi{x}}{\mathrm{2}}\right)}{\left({x}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{2}} }\:{dx}\:. \\ $$

Question Number 52683    Answers: 0   Comments: 3

let f(λ) =∫_(−∞) ^(+∞) ((sin(λx))/((x^2 +2λx +1)^2 ))dx with ∣λ∣<1 1) find the value of f(λ) 2) calculate ∫_(−∞) ^(+∞) ((sin((x/(2 ))))/((x^2 +x+1)^2 ))dx 3) find A(θ) =∫_(−∞) ^(+∞) ((sin((cosθ)x))/((x^2 +2cosθ x +1)^2 )) that we suppose 0<θ<(π/2)

$${let}\:{f}\left(\lambda\right)\:=\int_{−\infty} ^{+\infty} \:\:\frac{{sin}\left(\lambda{x}\right)}{\left({x}^{\mathrm{2}} \:+\mathrm{2}\lambda{x}\:+\mathrm{1}\right)^{\mathrm{2}} }{dx}\:\:\:\:{with}\:\mid\lambda\mid<\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{value}\:{of}\:{f}\left(\lambda\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{sin}\left(\frac{{x}}{\mathrm{2}\:}\right)}{\left({x}^{\mathrm{2}} \:\:+{x}+\mathrm{1}\right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{3}\right)\:{find}\:\:{A}\left(\theta\right)\:=\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{sin}\left(\left({cos}\theta\right){x}\right)}{\left({x}^{\mathrm{2}} \:+\mathrm{2}{cos}\theta\:{x}\:+\mathrm{1}\right)^{\mathrm{2}} }\:\:{that}\:{we}\:{suppose}\:\mathrm{0}<\theta<\frac{\pi}{\mathrm{2}} \\ $$

Question Number 52680    Answers: 0   Comments: 1

let f_n (x)=((sin(nx))/n^3 ) and f(x)=Σ_(n=1) ^∞ f_n (x) calculate ∫_0 ^π f(x)dx .

$${let}\:{f}_{{n}} \left({x}\right)=\frac{{sin}\left({nx}\right)}{{n}^{\mathrm{3}} }\:\:\:{and}\:{f}\left({x}\right)=\sum_{{n}=\mathrm{1}} ^{\infty} \:{f}_{{n}} \left({x}\right) \\ $$$${calculate}\:\int_{\mathrm{0}} ^{\pi} \:{f}\left({x}\right){dx}\:. \\ $$

Question Number 52667    Answers: 1   Comments: 0

∫((x^4 +1)/(x^2 (√(x^4 −1)))) dx

$$\int\frac{{x}^{\mathrm{4}} +\mathrm{1}}{{x}^{\mathrm{2}} \sqrt{{x}^{\mathrm{4}} −\mathrm{1}}}\:{dx} \\ $$

Question Number 52649    Answers: 0   Comments: 2

∫ ((4x^2 + 3)/((x^2 + x + 1)^2 )) dx

$$\int\:\frac{\mathrm{4x}^{\mathrm{2}} \:+\:\mathrm{3}}{\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{x}\:+\:\mathrm{1}\right)^{\mathrm{2}} }\:\mathrm{dx} \\ $$

Question Number 52550    Answers: 1   Comments: 1

∫_0 ^( ∞) (x/(e^x − 1)) dx

$$\:\:\int_{\mathrm{0}} ^{\:\infty} \:\:\:\frac{\mathrm{x}}{\mathrm{e}^{\mathrm{x}} \:−\:\mathrm{1}}\:\:\mathrm{dx}\: \\ $$

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