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IntegrationQuestion and Answers: Page 24

Question Number 185677 Answers: 1 Comments: 0

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Question Number 185455 Answers: 1 Comments: 0

∫ ((x^3 .e^x^2 )/((x^2 +1)^2 )) dx =?

$$\:\int\:\frac{{x}^{\mathrm{3}} .{e}^{{x}^{\mathrm{2}} } }{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} }\:{dx}\:=? \\ $$

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I_1 =∫_0 ^∞ (((√(x+(√(x^2 +1))))/( (√(x^2 +1))))−((√2)/( (√x))))dx=? I_2 =∫_0 ^∞ (((√(x^2 +1))/( (√(x+(√(x^2 +1))))))−((√x)/( (√2))))dx=?

$${I}_{\mathrm{1}} =\underset{\mathrm{0}} {\overset{\infty} {\int}}\left(\frac{\sqrt{{x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}}−\frac{\sqrt{\mathrm{2}}}{\:\sqrt{{x}}}\right){dx}=? \\ $$$${I}_{\mathrm{2}} =\underset{\mathrm{0}} {\overset{\infty} {\int}}\left(\frac{\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}}{\:\sqrt{{x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}}}−\frac{\sqrt{{x}}}{\:\sqrt{\mathrm{2}}}\right){dx}=? \\ $$

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∫_0 ^(2nπ) max(sin x, sin^(−1) (sin x)) dx =? [ n∈ I ]

$$\:\underset{\mathrm{0}} {\overset{\mathrm{2}{n}\pi} {\int}}\:{max}\left(\mathrm{sin}\:{x},\:\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{sin}\:{x}\right)\right)\:{dx}\:=?\: \\ $$$$\:\left[\:{n}\in\:{I}\:\right]\: \\ $$

Question Number 184112 Answers: 1 Comments: 1

∫ (dx/((x−1)^(5/6) (x+2)^(7/6) )) =?

$$\:\:\int\:\frac{{dx}}{\left({x}−\mathrm{1}\right)^{\mathrm{5}/\mathrm{6}} \left({x}+\mathrm{2}\right)^{\mathrm{7}/\mathrm{6}} }\:=? \\ $$

Question Number 184061 Answers: 1 Comments: 0

∫((sin^(−1) (√x)−cos^(−1) (√x))/(sin^(−1) (√x)+cos^(−1) (√x)))dx=?

$$\int\frac{{sin}^{−\mathrm{1}} \sqrt{{x}}−{cos}^{−\mathrm{1}} \sqrt{{x}}}{{sin}^{−\mathrm{1}} \sqrt{{x}}+{cos}^{−\mathrm{1}} \sqrt{{x}}}{dx}=? \\ $$

Question Number 183977 Answers: 3 Comments: 0

∫_0 ^( ∞) (dx/(x^8 +x^4 +1)) =?

$$\:\int_{\mathrm{0}} ^{\:\infty} \:\frac{{dx}}{{x}^{\mathrm{8}} +{x}^{\mathrm{4}} +\mathrm{1}}\:=?\: \\ $$

Question Number 183976 Answers: 1 Comments: 0

∫ ((sin x−(√(1+sin x)))/(cos x−(√(1+cos x)))) dx =?

$$\:\:\int\:\frac{\mathrm{sin}\:{x}−\sqrt{\mathrm{1}+\mathrm{sin}\:{x}}}{\mathrm{cos}\:{x}−\sqrt{\mathrm{1}+\mathrm{cos}\:{x}}}\:{dx}\:=? \\ $$

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∫ ((√(x+(√(x^2 +25))))/x) dx =?

$$\:\:\int\:\frac{\sqrt{{x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{25}}}}{{x}}\:{dx}\:=? \\ $$

Question Number 183761 Answers: 1 Comments: 0

Solve the differential equation for the function given by U(x,t). { (((∂U/∂t) = 2(∂^2 U/∂x^2 ) , 0 < x < π)),((U(0,t) = 0, U(π,t) = 0, t > 0)) :} U(x,0) = 25x

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{differential}\:\mathrm{equation}\:\mathrm{for}\:\mathrm{the}\:\mathrm{function} \\ $$$$\mathrm{given}\:\mathrm{by}\:{U}\left({x},{t}\right). \\ $$$$\begin{cases}{\frac{\partial{U}}{\partial{t}}\:=\:\mathrm{2}\frac{\partial^{\mathrm{2}} {U}}{\partial{x}^{\mathrm{2}} }\:,\:\mathrm{0}\:<\:{x}\:<\:\pi}\\{{U}\left(\mathrm{0},{t}\right)\:=\:\mathrm{0},\:{U}\left(\pi,{t}\right)\:=\:\mathrm{0},\:{t}\:>\:\mathrm{0}}\end{cases} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{U}\left({x},\mathrm{0}\right)\:=\:\mathrm{25}{x} \\ $$

Question Number 183794 Answers: 0 Comments: 1

If ∫_0 ^π ((cos x)/((x+2)^2 )) dx= T then ∫_0 ^π ((sin 2x)/(x+1)) dx = ?

$$\:{If}\:\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\frac{\mathrm{cos}\:{x}}{\left({x}+\mathrm{2}\right)^{\mathrm{2}} }\:{dx}=\:{T} \\ $$$$\:{then}\:\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\frac{\mathrm{sin}\:\mathrm{2}{x}}{{x}+\mathrm{1}}\:{dx}\:=\:?\: \\ $$

Question Number 183709 Answers: 0 Comments: 1

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