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Question Number 63273 Answers: 0 Comments: 1

let F(x) =∫_x^2 ^x^3 ((sin(t))/(t+x)) dt 1) calculate lim_(x→0) F(x) and lim_(x→+∞) F(x) 2)calculste lim_(x→0) F^′ (x) and lim_(x→+∞) F^′ (x)

$${let}\:{F}\left({x}\right)\:=\int_{{x}^{\mathrm{2}} } ^{{x}^{\mathrm{3}} } \:\:\:\:\:\frac{{sin}\left({t}\right)}{{t}+{x}}\:{dt} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:{F}\left({x}\right)\:{and}\:{lim}_{{x}\rightarrow+\infty} {F}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){calculste}\:{lim}_{{x}\rightarrow\mathrm{0}} \:{F}^{'} \left({x}\right)\:{and}\:{lim}_{{x}\rightarrow+\infty} \:{F}^{'} \left({x}\right) \\ $$

Question Number 63261 Answers: 0 Comments: 6

∫x tan(x) dx

$$\int{x}\:{tan}\left({x}\right)\:{dx} \\ $$

Question Number 63232 Answers: 0 Comments: 2

let B(x,y) =∫_0 ^1 (1−t)^(x−1) t^(y−1) dt 1) study the convergence of B(x,y) 1) prove that B(x,y)=B(y,x) prove that B(x,y) =∫_0 ^∞ (t^(x−1) /((1+t)^(x+y) )) dt 2) prove that B(x,y) =((Γ(x).Γ(y))/(Γ(x+y))) 3) prove that Γ(x).Γ(1−x) =(π/(sin(πx))) for allx ∈]0,1[

$${let}\:{B}\left({x},{y}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{t}\right)^{{x}−\mathrm{1}} {t}^{{y}−\mathrm{1}} \:{dt} \\ $$$$\left.\mathrm{1}\right)\:{study}\:{the}\:{convergence}\:{of}\:{B}\left({x},{y}\right) \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{B}\left({x},{y}\right)={B}\left({y},{x}\right) \\ $$$${prove}\:{that}\:{B}\left({x},{y}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}^{{x}−\mathrm{1}} }{\left(\mathrm{1}+{t}\right)^{{x}+{y}} }\:{dt} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:{B}\left({x},{y}\right)\:=\frac{\Gamma\left({x}\right).\Gamma\left({y}\right)}{\Gamma\left({x}+{y}\right)} \\ $$$$\left.\mathrm{3}\left.\right)\:{prove}\:{that}\:\Gamma\left({x}\right).\Gamma\left(\mathrm{1}−{x}\right)\:=\frac{\pi}{{sin}\left(\pi{x}\right)}\:\:\:{for}\:{allx}\:\in\right]\mathrm{0},\mathrm{1}\left[\right. \\ $$

Question Number 63251 Answers: 0 Comments: 0

∫_( 0) ^( (π/2)) sin^(−1) (m cosθ) dθ

$$\int_{\:\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \:\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{m}\:\mathrm{cos}\theta\right)\:\mathrm{d}\theta \\ $$

Question Number 63214 Answers: 0 Comments: 1

calculate ∫_0 ^∞ x e^(−(x^2 /a^2 )) sin(bx)dx with a>0 and b>0

$${calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:{x}\:{e}^{−\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }} \:\:{sin}\left({bx}\right){dx}\:\:{with}\:\:{a}>\mathrm{0}\:{and}\:{b}>\mathrm{0} \\ $$

Question Number 63268 Answers: 0 Comments: 0

Question Number 63292 Answers: 0 Comments: 4

∫_0 ^1 ∫_0 ^1 (dy/(1+y(x^2 −x))) dx

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

Question Number 63139 Answers: 0 Comments: 3

Σ_(n = 1) ^m ((log n)/n^(3/2) )

$$\underset{\mathrm{n}\:=\:\mathrm{1}} {\overset{\mathrm{m}} {\sum}}\:\frac{\mathrm{log}\:\mathrm{n}}{\mathrm{n}^{\mathrm{3}/\mathrm{2}} } \\ $$

Question Number 63117 Answers: 1 Comments: 1

∫((cos x)/(2+3sin x+sin^2 x))dx

$$\int\frac{\mathrm{cos}\:{x}}{\mathrm{2}+\mathrm{3sin}\:{x}+\mathrm{sin}\:^{\mathrm{2}} {x}}{dx} \\ $$

Question Number 63116 Answers: 1 Comments: 1

∫((1+×)/(√(1+×^2 )))dx

$$\int\frac{\mathrm{1}+×}{\sqrt{\mathrm{1}+×^{\mathrm{2}} }}{dx} \\ $$

Question Number 63089 Answers: 0 Comments: 3

find the value of ∫_0 ^(π/2) (dx/(1+(tanx)^(√2) )) .

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{dx}}{\mathrm{1}+\left({tanx}\right)^{\sqrt{\mathrm{2}}} }\:. \\ $$

Question Number 63084 Answers: 0 Comments: 1

let f(z) =((cos(3z))/z^2 ) calculate Res(f,0) .

$${let}\:{f}\left({z}\right)\:=\frac{{cos}\left(\mathrm{3}{z}\right)}{{z}^{\mathrm{2}} } \\ $$$${calculate}\:{Res}\left({f},\mathrm{0}\right)\:. \\ $$

Question Number 63080 Answers: 0 Comments: 0

∫((√((sinx)/x^3 ))/x^3 )dx

$$\:\int\frac{\sqrt{\frac{{sinx}}{{x}^{\mathrm{3}} }}}{{x}^{\mathrm{3}} }{dx} \\ $$

Question Number 63079 Answers: 0 Comments: 1

let f(z) =((sin(2z))/z^n ) with n integr natural calculate Res(f,0)

$${let}\:{f}\left({z}\right)\:=\frac{{sin}\left(\mathrm{2}{z}\right)}{{z}^{{n}} }\:\:\:\:{with}\:{n}\:{integr}\:{natural}\: \\ $$$${calculate}\:{Res}\left({f},\mathrm{0}\right) \\ $$

Question Number 63034 Answers: 0 Comments: 0

calculate ∫_0 ^(π/2) (ln(cosx))^2 dx

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

Question Number 63033 Answers: 0 Comments: 1

calculate ∫_0 ^∞ ((sin^2 (x))/(x^2 (1+x^2 )))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sin}^{\mathrm{2}} \left({x}\right)}{{x}^{\mathrm{2}} \left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{dx}\: \\ $$

Question Number 63032 Answers: 0 Comments: 1

let f(z) =(1/(sin(πz))) calculate Res(f,n) with n integr

$${let}\:{f}\left({z}\right)\:=\frac{\mathrm{1}}{{sin}\left(\pi{z}\right)}\:\:{calculate}\:{Res}\left({f},{n}\right)\:{with}\:{n}\:{integr} \\ $$

Question Number 63031 Answers: 0 Comments: 2

let f(z) =((sin(z))/z^2 ) calculate Res(f,0)

$${let}\:{f}\left({z}\right)\:=\frac{{sin}\left({z}\right)}{{z}^{\mathrm{2}} }\:\:{calculate}\:{Res}\left({f},\mathrm{0}\right) \\ $$

Question Number 63026 Answers: 0 Comments: 0

calculate ∫_0 ^π ((sin(2x))/(2cosx −3sinx))dx

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

Question Number 63023 Answers: 0 Comments: 1

calculate ∫_(−∞) ^(+∞) ((x^2 −3)/(x^4 +x^2 +1))dx .

$${calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{x}^{\mathrm{2}} −\mathrm{3}}{{x}^{\mathrm{4}} \:+{x}^{\mathrm{2}} \:+\mathrm{1}}{dx}\:. \\ $$

Question Number 62997 Answers: 0 Comments: 1

∫((ln(1+xsin^2 (x)))/(sin^2 (x))) dx

$$\int\frac{{ln}\left(\mathrm{1}+{xsin}^{\mathrm{2}} \left({x}\right)\right)}{{sin}^{\mathrm{2}} \left({x}\right)}\:{dx} \\ $$

Question Number 62995 Answers: 0 Comments: 0

Question Number 62981 Answers: 1 Comments: 1

Question Number 62937 Answers: 1 Comments: 3

∫_0 ^( x) (1/(1+x^2 )) dx

$$\int_{\mathrm{0}} ^{\:\:\mathrm{x}} \frac{\mathrm{1}}{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx} \\ $$

Question Number 62930 Answers: 0 Comments: 0

find the value ∫_0 ^1 x^(√x) dx (study first the convergence)

$${find}\:{the}\:{value}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{\sqrt{{x}}} {dx}\:\left({study}\:{first}\:{the}\:{convergence}\right) \\ $$

Question Number 62912 Answers: 1 Comments: 0

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