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IntegrationQuestion and Answers: Page 20

Question Number 195846 Answers: 1 Comments: 0

{ (( Ω_1 = ∫_0 ^( (π/2)) (( x^( 2) )/(sin^( 2) (x))) dx )),(( ⇒ (Ω_1 /Ω_( 2) ) = ? )),(( Ω_( 2) = ∫_0 ^( (π/2)) (x/(tan(x))) dx)) :}

$$ \\ $$$$\:\:\:\:\:\:\begin{cases}{\:\:\:\Omega_{\mathrm{1}} \:=\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{\:{x}^{\:\mathrm{2}} }{{sin}^{\:\mathrm{2}} \left({x}\right)}\:{dx}\:}\\{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\:\:\:\frac{\Omega_{\mathrm{1}} }{\Omega_{\:\mathrm{2}} }\:=\:?\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:}\\{\:\:\Omega_{\:\mathrm{2}} =\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{{x}}{{tan}\left({x}\right)}\:{dx}}\end{cases} \\ $$$$ \\ $$

Question Number 195803 Answers: 0 Comments: 0

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

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

Question Number 195468 Answers: 1 Comments: 0

∫((x^5 +x^2 )/((x^6 +2x^3 )^7 ))dx

$$\int\frac{{x}^{\mathrm{5}} +{x}^{\mathrm{2}} }{\left({x}^{\mathrm{6}} +\mathrm{2}{x}^{\mathrm{3}} \right)^{\mathrm{7}} }{dx} \\ $$

Question Number 195416 Answers: 1 Comments: 0

Question Number 200299 Answers: 1 Comments: 0

Question Number 195224 Answers: 2 Comments: 0

∫_1 ^e 0 dx = ??

$$\:\:\:\:\:\:\:\int_{\mathrm{1}} ^{\mathrm{e}} \mathrm{0}\:{dx}\:=\:?? \\ $$

Question Number 195126 Answers: 1 Comments: 0

Soit f_n (x)=2^(n+1) [(((1/2^n )cotan((x/2^n ))−cotanx)/(sin((x/2^n ))))] Calculer lim_(x→0) f_n (x) et lim_(n→+∞) ((f_n (x))/2^(2n+2) )

$$\mathrm{Soit}\:{f}_{{n}} \left({x}\right)=\mathrm{2}^{{n}+\mathrm{1}} \left[\frac{\frac{\mathrm{1}}{\mathrm{2}^{{n}} }{cotan}\left(\frac{{x}}{\mathrm{2}^{{n}} }\right)−{cotanx}}{{sin}\left(\frac{{x}}{\mathrm{2}^{{n}} }\right)}\right] \\ $$$${Calculer}\:\underset{{x}\rightarrow\mathrm{0}} {{lim}f}_{{n}} \left({x}\right)\:{et}\:\underset{{n}\rightarrow+\infty} {{lim}}\:\frac{{f}_{{n}} \left({x}\right)}{\mathrm{2}^{\mathrm{2}{n}+\mathrm{2}} } \\ $$

Question Number 195082 Answers: 0 Comments: 0

Question Number 194963 Answers: 2 Comments: 0

Question Number 194975 Answers: 0 Comments: 6

∫(1/(x^5 +1))dx

$$\int\frac{\mathrm{1}}{\boldsymbol{{x}}^{\mathrm{5}} +\mathrm{1}}\boldsymbol{{dx}} \\ $$

Question Number 194930 Answers: 0 Comments: 1

∫_0 ^1 x^(−x) dx=?

$$\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{−{x}} {dx}=? \\ $$

Question Number 194850 Answers: 0 Comments: 0

Question Number 194785 Answers: 0 Comments: 0

∫∫ x^2 +y^2 dxdy (D=x^4 +y^4 ≤1)

$$\int\int\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \:{dxdy}\:\left({D}={x}^{\mathrm{4}} +{y}^{\mathrm{4}} \leqslant\mathrm{1}\right) \\ $$

Question Number 194759 Answers: 1 Comments: 1

∫_0 ^( 1) ∫_0 ^( 1) (((1+x^2 )/(1+x^2 +y^2 ))) dxdy

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

Question Number 194591 Answers: 0 Comments: 0

∫_(−2) ^2 ∫_(2x^2 ) ^8 ∫_(−(√((1/2)y−x^2 ))) ^(√((1/2)y−x^2 )) ((√(3x^2 +3z^2 )) )dzdydx

$$\:\:\:\underset{−\mathrm{2}} {\overset{\mathrm{2}} {\int}}\:\underset{\mathrm{2x}^{\mathrm{2}} } {\overset{\mathrm{8}} {\int}}\:\:\underset{−\sqrt{\frac{\mathrm{1}}{\mathrm{2}}\mathrm{y}−\mathrm{x}^{\mathrm{2}} }} {\overset{\sqrt{\frac{\mathrm{1}}{\mathrm{2}}\mathrm{y}−\mathrm{x}^{\mathrm{2}} }} {\int}}\:\left(\sqrt{\mathrm{3x}^{\mathrm{2}} +\mathrm{3z}^{\mathrm{2}} }\:\right)\mathrm{dzdydx} \\ $$

Question Number 194548 Answers: 2 Comments: 1

Question Number 194515 Answers: 1 Comments: 0

∣∫f(x)dx∣=∫∣f(x)∣dx

$$\mid\int{f}\left({x}\right){dx}\mid=\int\mid{f}\left({x}\right)\mid{dx} \\ $$

Question Number 194456 Answers: 2 Comments: 0

$$\:\:\:\:\:\:\cancel{ } \\ $$

Question Number 194412 Answers: 1 Comments: 0

Question Number 194388 Answers: 0 Comments: 0

∫(((1+x^2 )sin x+4cos x)/(2(1+x^2 )^3 )) dx

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

Question Number 193982 Answers: 3 Comments: 0

∫ ((6x^3 +9x^2 +15x+6)/( (√(x^2 +x+1)))) dx =?

$$\:\:\:\:\:\int\:\frac{\mathrm{6x}^{\mathrm{3}} +\mathrm{9x}^{\mathrm{2}} +\mathrm{15x}+\mathrm{6}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}}}\:\mathrm{dx}\:=? \\ $$

Question Number 193852 Answers: 3 Comments: 0

Question Number 193538 Answers: 2 Comments: 0

∫ (dx/(x^6 +1)) =?

$$\:\:\:\int\:\frac{\mathrm{dx}}{\mathrm{x}^{\mathrm{6}} +\mathrm{1}}\:=? \\ $$

Question Number 193526 Answers: 2 Comments: 0

Question Number 193512 Answers: 1 Comments: 0

Question Number 193439 Answers: 1 Comments: 2

∫^(π/2) _( 0) (((tanx))^(1/3) /((sinx+cosx)^2 ))dx

$$\underset{\:\:\mathrm{0}} {\int}^{\pi/\mathrm{2}} \frac{\sqrt[{\mathrm{3}}]{{tanx}}}{\left({sinx}+{cosx}\right)^{\mathrm{2}} }{dx} \\ $$

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