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IntegrationQuestion and Answers: Page 194

Question Number 84879 Answers: 2 Comments: 1

e^(∫((2dx)/(xlnx)))

$$\mathrm{e}^{\int\frac{\mathrm{2dx}}{\mathrm{xlnx}}} \\ $$

Question Number 84859 Answers: 0 Comments: 0

show that lim_(n→∞) ∫_0 ^1 ...∫_0 ^1 (n/(x_1 +x_2 +x_3 +...+x_n ))dx_1 dx_2 ...dx_n =2

$${show}\:{that} \\ $$$$\underset{{n}\rightarrow\infty} {{lim}}\int_{\mathrm{0}} ^{\mathrm{1}} ...\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{n}}{{x}_{\mathrm{1}} +{x}_{\mathrm{2}} +{x}_{\mathrm{3}} +...+{x}_{{n}} }{dx}_{\mathrm{1}} {dx}_{\mathrm{2}} ...{dx}_{{n}} =\mathrm{2}\: \\ $$

Question Number 84843 Answers: 0 Comments: 2

∫((sin(7x))/(cos(3x))) dx

$$\int\frac{{sin}\left(\mathrm{7}{x}\right)}{{cos}\left(\mathrm{3}{x}\right)}\:{dx} \\ $$

Question Number 84810 Answers: 0 Comments: 1

∫_0 ^π ln(((1+b cos(x))/(1+a sin(x)))) dx −1<a<b<1

$$\int_{\mathrm{0}} ^{\pi} {ln}\left(\frac{\mathrm{1}+{b}\:{cos}\left({x}\right)}{\mathrm{1}+{a}\:{sin}\left({x}\right)}\right)\:{dx} \\ $$$$−\mathrm{1}<{a}<{b}<\mathrm{1} \\ $$

Question Number 84809 Answers: 1 Comments: 1

∫(x/((x^2 +1)^(3/2) arctan(x))) dx

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

Question Number 84766 Answers: 1 Comments: 2

∫ (dx/((16+9sin x)^2 ))

$$\int\:\frac{\mathrm{dx}}{\left(\mathrm{16}+\mathrm{9sin}\:\mathrm{x}\right)^{\mathrm{2}} } \\ $$$$ \\ $$

Question Number 84759 Answers: 1 Comments: 0

calculate ∫_(−∞) ^(+∞) ((arctan(2x^2 ))/(1+x^2 ))dx

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

Question Number 84733 Answers: 1 Comments: 0

∫((x^4 +1)/(x^6 +1))dx

$$\int\frac{{x}^{\mathrm{4}} +\mathrm{1}}{{x}^{\mathrm{6}} +\mathrm{1}}{dx} \\ $$

Question Number 84726 Answers: 0 Comments: 0

prove that ∫_0 ^1 (y^y )^((y^y )^((y^y )^.^.^. ) ) dy=(π^2 /(12))

$${prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \left({y}^{{y}} \right)^{\left({y}^{{y}} \right)^{\left({y}^{{y}} \right)^{.^{.^{.} } } } } {dy}=\frac{\pi^{\mathrm{2}} }{\mathrm{12}} \\ $$

Question Number 84713 Answers: 2 Comments: 2

∫ _0 ^( 1) cos^(−1) (x^2 −x+1) dx ?

$$\int\overset{\:\:\mathrm{1}} {\:}_{\mathrm{0}} \mathrm{cos}^{−\mathrm{1}} \left(\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{1}\right)\:\mathrm{dx}\:? \\ $$

Question Number 84702 Answers: 0 Comments: 3

Question Number 84709 Answers: 1 Comments: 0

∫((tan(x)))^(1/3) dx

$$\int\sqrt[{\mathrm{3}}]{{tan}\left({x}\right)}\:{dx} \\ $$

Question Number 84708 Answers: 1 Comments: 0

Question Number 84693 Answers: 1 Comments: 1

Question Number 84685 Answers: 0 Comments: 0

∫(dx/(x(2−e^x )))

$$\int\frac{{dx}}{{x}\left(\mathrm{2}−{e}^{{x}} \right)} \\ $$

Question Number 84681 Answers: 1 Comments: 1

Question Number 84666 Answers: 1 Comments: 2

∫ x sin^(−1) (x) dx

$$\int\:{x}\:\mathrm{sin}^{−\mathrm{1}} \left({x}\right)\:{dx}\: \\ $$

Question Number 84580 Answers: 0 Comments: 0

find A_λ =∫ ((x+1)/(x+3))(√((λ+x)/(λ−x)))dx

$${find}\:{A}_{\lambda} =\int\:\frac{{x}+\mathrm{1}}{{x}+\mathrm{3}}\sqrt{\frac{\lambda+{x}}{\lambda−{x}}}{dx} \\ $$

Question Number 84578 Answers: 0 Comments: 4

calculate ∫_0 ^(π/4) (dx/((cosx +3sinx)^2 ))

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{dx}}{\left({cosx}\:+\mathrm{3}{sinx}\right)^{\mathrm{2}} } \\ $$

Question Number 84577 Answers: 0 Comments: 2

calculate ∫ (dx/(cosx +cos(2x)+cos(3x)))

$${calculate}\:\int\:\:\:\:\frac{{dx}}{{cosx}\:+{cos}\left(\mathrm{2}{x}\right)+{cos}\left(\mathrm{3}{x}\right)} \\ $$

Question Number 84574 Answers: 0 Comments: 1

calculate I_n =∫_0 ^1 sin(narcsinx)dx

$${calculate}\:\:{I}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:{sin}\left({narcsinx}\right){dx} \\ $$

Question Number 84570 Answers: 1 Comments: 1

calculate ∫_1 ^(+∞) ((arctan((3/x)))/x^2 )dx

$${calculate}\:\int_{\mathrm{1}} ^{+\infty} \:\frac{{arctan}\left(\frac{\mathrm{3}}{{x}}\right)}{{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 84569 Answers: 1 Comments: 0

find ∫ (dx/(1+tan^4 x))

$${find}\:\:\int\:\:\:\:\frac{{dx}}{\mathrm{1}+{tan}^{\mathrm{4}} {x}} \\ $$

Question Number 84566 Answers: 0 Comments: 0

Question Number 84561 Answers: 1 Comments: 0

∫_0 ^∞ ∫_0 ^∞ ((cos(x−y)−cos(x))/(xy))dx dy

$$\int_{\mathrm{0}} ^{\infty} \int_{\mathrm{0}} ^{\infty} \frac{{cos}\left({x}−{y}\right)−{cos}\left({x}\right)}{{xy}}{dx}\:{dy} \\ $$

Question Number 84557 Answers: 0 Comments: 1

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