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Question Number 29441    Answers: 0   Comments: 1

find ∫ (x^2 /((2−x^2 )(√(1−x^2 ))))dx

$${find}\:\int\:\:\frac{{x}^{\mathrm{2}} }{\left(\mathrm{2}−{x}^{\mathrm{2}} \right)\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}{dx} \\ $$

Question Number 29440    Answers: 0   Comments: 0

find ∫_0 ^π ((cosx)/((2+cosx)(3+cosx)))dx

$${find}\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\:\:\frac{{cosx}}{\left(\mathrm{2}+{cosx}\right)\left(\mathrm{3}+{cosx}\right)}{dx} \\ $$

Question Number 29439    Answers: 1   Comments: 0

find ∫_0 ^π (dx/(2+cosx)) .

$${find}\:\int_{\mathrm{0}} ^{\pi} \:\frac{{dx}}{\mathrm{2}+{cosx}}\:. \\ $$

Question Number 29384    Answers: 0   Comments: 3

Please can it be proven by another means that ∫tan^2 xdx=tanx+x +c

$${Please}\:{can}\:{it}\:{be}\:{proven}\:{by}\:{another} \\ $$$${means}\:{that}\: \\ $$$$ \\ $$$$\:\:\:\:\:\int\mathrm{tan}\:^{\mathrm{2}} {xdx}={tanx}+{x}\:+{c} \\ $$

Question Number 29423    Answers: 0   Comments: 1

Question Number 29311    Answers: 0   Comments: 0

∫(2x^3 −3x^2 +3x−1)^(1/5) dx and limit is from 0 to 1

$$\int\left(\mathrm{2}{x}^{\mathrm{3}} −\mathrm{3}{x}^{\mathrm{2}} +\mathrm{3}{x}−\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{5}}} {dx}\:{and}\:{limit}\:{is}\:{from}\:\mathrm{0}\:{to}\:\mathrm{1} \\ $$

Question Number 29202    Answers: 1   Comments: 0

Find area between by y=1 and y=((1−x^2 )/(1+x^2 )) .

$${Find}\:{area}\:{between}\:{by}\:{y}=\mathrm{1}\:\:{and} \\ $$$${y}=\frac{\mathrm{1}−{x}^{\mathrm{2}} }{\mathrm{1}+{x}^{\mathrm{2}} }\:. \\ $$

Question Number 29162    Answers: 0   Comments: 1

find find I= ∫_1 ^3 ((∣x−2∣)/((x^2 −4x)^2 ))dx .

$${find}\:\:{find}\:{I}=\:\int_{\mathrm{1}} ^{\mathrm{3}} \:\:\:\:\frac{\mid{x}−\mathrm{2}\mid}{\left({x}^{\mathrm{2}} −\mathrm{4}{x}\right)^{\mathrm{2}} }{dx}\:. \\ $$

Question Number 29105    Answers: 0   Comments: 2

Show that: ∫_(−1) ^( 1) (dx/(5 cosh(x) + 13 sinh(x))) = (1/2) log_e (((15e − 10)/(3e + 2)))

$$\mathrm{Show}\:\mathrm{that}:\:\:\:\int_{−\mathrm{1}} ^{\:\:\:\mathrm{1}} \:\:\:\:\:\:\:\frac{\mathrm{dx}}{\mathrm{5}\:\mathrm{cosh}\left(\mathrm{x}\right)\:+\:\mathrm{13}\:\mathrm{sinh}\left(\mathrm{x}\right)}\:\:=\:\:\frac{\mathrm{1}}{\mathrm{2}}\:\mathrm{log}_{\mathrm{e}} \left(\frac{\mathrm{15e}\:−\:\mathrm{10}}{\mathrm{3e}\:+\:\mathrm{2}}\right)\: \\ $$

Question Number 29079    Answers: 0   Comments: 0

let give w(x)= ∫_0 ^1 ((arcsin(x(1+t^2 )))/(1+t^2 ))dt find w(x).

$${let}\:{give}\:{w}\left({x}\right)=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{arcsin}\left({x}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)\right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}\:\:{find}\:{w}\left({x}\right). \\ $$

Question Number 29078    Answers: 0   Comments: 2

let give h(x)= ∫_0 ^1 ((arctan(xt))/(1+t^2 )) find h(x) .

$${let}\:{give}\:\:{h}\left({x}\right)=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{arctan}\left({xt}\right)}{\mathrm{1}+{t}^{\mathrm{2}} }\:\:{find}\:{h}\left({x}\right)\:. \\ $$

Question Number 29077    Answers: 0   Comments: 1

let give g(x)=∫_0 ^∞ ((arctan(x(1+t^2 )))/(1+t^2 ))dt find a simple form of g^′ (x) without integral.

$${let}\:{give}\:{g}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left({x}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)\right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}\:\:{find}\:{a}\:{simple} \\ $$$${form}\:{of}\:\:{g}^{'} \left({x}\right)\:{without}\:{integral}. \\ $$

Question Number 29076    Answers: 0   Comments: 1

let give f(x)= ∫_0 ^1 ((arctan(x(1+t^2 )))/(1+t^2 ))dt find asimple form of f(x) without integral.

$${let}\:{give}\:{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{arctan}\left({x}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)\right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}\:\:{find}\:{asimple} \\ $$$${form}\:{of}\:{f}\left({x}\right)\:{without}\:{integral}. \\ $$

Question Number 29043    Answers: 0   Comments: 1

∫tan^− (1−sinx/1+sinx) dx

$$\int\mathrm{tan}^{−} \left(\mathrm{1}−\mathrm{sinx}/\mathrm{1}+\mathrm{sinx}\right)\:\mathrm{dx} \\ $$

Question Number 29038    Answers: 0   Comments: 1

find ∫_(−∞) ^(+∞) ((cos(at))/(1+t^4 ))dt.

$${find}\:\:\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{cos}\left({at}\right)}{\mathrm{1}+{t}^{\mathrm{4}} }{dt}. \\ $$

Question Number 29028    Answers: 0   Comments: 0

for t>0 and f(t)= (4πt)^(−(n/2)) e^(−(x^2 /(4t))) prove that ∫_R f_t (x)dx=1 ∀t>0.

$${for}\:{t}>\mathrm{0}\:\:{and}\:{f}\left({t}\right)=\:\left(\mathrm{4}\pi{t}\right)^{−\frac{{n}}{\mathrm{2}}} \:\:{e}^{−\frac{{x}^{\mathrm{2}} }{\mathrm{4}{t}}} \:\:\:{prove}\:{that} \\ $$$$\int_{{R}} {f}_{{t}} \left({x}\right){dx}=\mathrm{1}\:\:\:\forall{t}>\mathrm{0}. \\ $$

Question Number 29027    Answers: 0   Comments: 0

find ∫∫_D e^(−y) sin(2xy)dxdy with D=[0,1]×[0,+∞[ then find the value of ∫_0 ^∞ ((sin^2 t)/t) e^(−t) dt .

$${find}\:\int\int_{{D}} \:{e}^{−{y}} {sin}\left(\mathrm{2}{xy}\right){dxdy}\:{with}\:{D}=\left[\mathrm{0},\mathrm{1}\right]×\left[\mathrm{0},+\infty\left[\right.\right. \\ $$$${then}\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\frac{{sin}^{\mathrm{2}} {t}}{{t}}\:{e}^{−{t}} {dt}\:\:. \\ $$

Question Number 29018    Answers: 0   Comments: 0

∫ (√(Σ_(n = 0) ^∞ [(−1)^n tan^(2n) (2x)])) dx

$$\int\:\sqrt{\underset{{n}\:=\:\mathrm{0}} {\overset{\infty} {\sum}}\left[\left(−\mathrm{1}\right)^{{n}} \:\mathrm{tan}^{\mathrm{2}{n}} \:\left(\mathrm{2}{x}\right)\right]}\:{dx} \\ $$

Question Number 29003    Answers: 1   Comments: 1

find ∫_0 ^∞ (dx/(1+x^3 )) .

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{3}} }\:. \\ $$

Question Number 29002    Answers: 0   Comments: 0

let give 0<p<1 calculate K(p)= ∫_(−∞) ^(+∞) (e^(pt) /(1+e^t ))dt.

$${let}\:{give}\:\mathrm{0}<{p}<\mathrm{1}\:{calculate}\:\:{K}\left({p}\right)=\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{e}^{{pt}} }{\mathrm{1}+{e}^{{t}} }{dt}. \\ $$

Question Number 29001    Answers: 0   Comments: 0

find the value of ∫_0 ^∞ ((cos(ξt))/(1+t^4 ))dt.

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{cos}\left(\xi{t}\right)}{\mathrm{1}+{t}^{\mathrm{4}} }{dt}. \\ $$

Question Number 29000    Answers: 0   Comments: 1

prove thst ∫_R (e^(iξx) /(1+x^2 ))dx= π e^(−∣ξ∣) .

$${prove}\:{thst}\:\:\:\:\int_{\mathbb{R}} \:\:\:\:\frac{{e}^{{i}\xi{x}} }{\mathrm{1}+{x}^{\mathrm{2}} }{dx}=\:\pi\:{e}^{−\mid\xi\mid} \:\:. \\ $$

Question Number 28999    Answers: 0   Comments: 1

prove that ∫_0 ^∞ (e^(−t) /(√t))dt= e^(i(π/4)) ∫_0 ^∞ (e^(−ix) /(√x))dx.

$${prove}\:{that}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{t}} }{\sqrt{{t}}}{dt}=\:{e}^{{i}\frac{\pi}{\mathrm{4}}} \:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{ix}} }{\sqrt{{x}}}{dx}. \\ $$

Question Number 28998    Answers: 0   Comments: 0

find ∫_γ (e^z /(z(z+1)))dz with γ={z∈C/ ∣z−1∣=2}

$${find}\:\int_{\gamma} \:\:\:\:\frac{{e}^{{z}} }{{z}\left({z}+\mathrm{1}\right)}{dz}\:{with}\:\gamma=\left\{{z}\in{C}/\:\mid{z}−\mathrm{1}\mid=\mathrm{2}\right\} \\ $$

Question Number 28997    Answers: 0   Comments: 1

find ∫_(−∞) ^(+∞) (dx/((1+x^2 )( 2+e^(ix) ))) .

$${find}\:\int_{−\infty} ^{+\infty} \:\:\:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\:\mathrm{2}+{e}^{{ix}} \right)}\:. \\ $$

Question Number 28996    Answers: 0   Comments: 0

find ∫_(−∞) ^(+∞) (x^2 /((x^2 +1)^2 (x^2 +2x+2)))dx.

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

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