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IntegrationQuestion and Answers: Page 204

Question Number 78273 Answers: 0 Comments: 1

let f(θ) =∫_0 ^(π/4) (dx/(1+sinθ sinx)) with 0<θ<(π/2) 1) explicite f(θ) 2) calculate ∫_0 ^(π/4) (dx/((1+sinθ sinx)^2 ))

$${let}\:{f}\left(\theta\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{dx}}{\mathrm{1}+{sin}\theta\:{sinx}} \\ $$$$ \\ $$$${with}\:\mathrm{0}<\theta<\frac{\pi}{\mathrm{2}} \\ $$$$\left.\mathrm{1}\right)\:{explicite}\:{f}\left(\theta\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{dx}}{\left(\mathrm{1}+{sin}\theta\:{sinx}\right)^{\mathrm{2}} } \\ $$

Question Number 78271 Answers: 0 Comments: 1

calculate A_θ =∫_0 ^(π/2) (dx/(2+cosθ sinx)) −π<θ<π

$${calculate}\:{A}_{\theta} \:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{dx}}{\mathrm{2}+{cos}\theta\:{sinx}} \\ $$$$−\pi<\theta<\pi \\ $$

Question Number 78270 Answers: 1 Comments: 1

find ∫_0 ^1 ((ln(1−x^2 )ln(x))/x^2 )dx prove first the convergence.

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}−{x}^{\mathrm{2}} \right){ln}\left({x}\right)}{{x}^{\mathrm{2}} }{dx} \\ $$$${prove}\:{first}\:{the}\:{convergence}. \\ $$

Question Number 78269 Answers: 1 Comments: 1

let f(a) =∫_0 ^∞ ((cos(ax))/(x^2 +a^2 ))dx with a>0 find ∫_1 ^2 f(a)da

$${let}\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\frac{{cos}\left({ax}\right)}{{x}^{\mathrm{2}} +{a}^{\mathrm{2}} }{dx}\:{with} \\ $$$${a}>\mathrm{0}\:\:{find}\:\:\:\int_{\mathrm{1}} ^{\mathrm{2}} {f}\left({a}\right){da} \\ $$

Question Number 78267 Answers: 0 Comments: 1

find ∫_(−∞) ^(+∞) ((x^2 −x)/(x^4 −x^2 +3))dx

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

Question Number 78266 Answers: 1 Comments: 2

calculate f(a)=∫_0 ^1 ln(1−ax^3 )dx with 0<a<1

$${calculate}\:{f}\left({a}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}−{ax}^{\mathrm{3}} \right){dx} \\ $$$${with}\:\mathrm{0}<{a}<\mathrm{1} \\ $$

Question Number 78264 Answers: 0 Comments: 4

let I =∫_0 ^π x cos^4 x dxand J=∫_0 ^π x sin^4 xdx 1) calculate I+J and I−J 2) find the values of I and J

$${let}\:{I}\:=\int_{\mathrm{0}} ^{\pi} {x}\:{cos}^{\mathrm{4}} {x}\:{dxand}\:{J}=\int_{\mathrm{0}} ^{\pi} {x}\:{sin}^{\mathrm{4}} {xdx} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{I}+{J}\:{and}\:{I}−{J} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\:{I}\:{and}\:{J} \\ $$

Question Number 78265 Answers: 1 Comments: 0

find ∫ ((sin^3 x)/(tan^5 x))dx

$${find}\:\int\:\:\frac{{sin}^{\mathrm{3}} {x}}{{tan}^{\mathrm{5}} {x}}{dx} \\ $$

Question Number 78251 Answers: 1 Comments: 0

∫((x+4)/(x−(x)^(1/3) )) dx

$$\int\frac{{x}+\mathrm{4}}{{x}−\sqrt[{\mathrm{3}}]{{x}}}\:{dx}\: \\ $$

Question Number 78198 Answers: 1 Comments: 4

Question Number 78163 Answers: 0 Comments: 2

∫ (√(1 + 3 sin(θ) + sin^2 (θ))) dθ

$$\int\:\sqrt{\mathrm{1}\:+\:\mathrm{3}\:\mathrm{sin}\left(\theta\right)\:+\:\mathrm{sin}^{\mathrm{2}} \left(\theta\right)}\:\:\mathrm{d}\theta \\ $$

Question Number 78135 Answers: 0 Comments: 0

explicit f(x)=∫_0 ^∞ ((arctan(xt))/(t^2 +x^2 ))dt with x>0

$${explicit}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left({xt}\right)}{{t}^{\mathrm{2}} +{x}^{\mathrm{2}} }{dt} \\ $$$${with}\:{x}>\mathrm{0} \\ $$

Question Number 78134 Answers: 0 Comments: 1

find the value of ∫_(−∞) ^(+∞) ((arctan(3x^2 ))/(x^2 +4))dx

$${find}\:{the}\:{value}\:{of}\:\:\int_{−\infty} ^{+\infty} \:\frac{{arctan}\left(\mathrm{3}{x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} +\mathrm{4}}{dx} \\ $$

Question Number 78254 Answers: 0 Comments: 0

∫_0 ^1 ((xln(ln((1/x))))/((x^2 −x+1)^2 ))dx i poste solution later!

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{xln}\left(\mathrm{ln}\left(\frac{\mathrm{1}}{\mathrm{x}}\right)\right)}{\left(\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} }\mathrm{dx} \\ $$$$\mathrm{i}\:\mathrm{poste}\:\mathrm{solution}\:\mathrm{later}! \\ $$

Question Number 78013 Answers: 1 Comments: 0

if : 30x^4 −((15)/8)= ∫_t ^x g(u)du find g(t).

$${if}\::\:\mathrm{30}{x}^{\mathrm{4}} −\frac{\mathrm{15}}{\mathrm{8}}=\:\underset{{t}} {\overset{{x}} {\int}}\:{g}\left({u}\right){du} \\ $$$${find}\:{g}\left({t}\right). \\ $$

Question Number 77995 Answers: 0 Comments: 3

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

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

Question Number 77987 Answers: 0 Comments: 0

Question Number 77962 Answers: 1 Comments: 2

∫(dx/(1+(tan(x))^(√2) )) dx

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

Question Number 77960 Answers: 0 Comments: 3

∫ ((2x^3 −1)/(x^4 +x)) dx?

$$\int\:\frac{\mathrm{2}{x}^{\mathrm{3}} −\mathrm{1}}{{x}^{\mathrm{4}} +{x}}\:{dx}? \\ $$

Question Number 77918 Answers: 1 Comments: 1

∫ _0 ^π e^(−2x) sin x dx ?

$$\int\underset{\mathrm{0}} {\overset{\pi} {\:}}\:{e}^{−\mathrm{2}{x}} \:\mathrm{sin}\:{x}\:{dx}\:?\: \\ $$

Question Number 77887 Answers: 1 Comments: 0

∫(e^(sinh(x)) /(cosh(x))) dx

$$\int\frac{{e}^{{sinh}\left({x}\right)} }{{cosh}\left({x}\right)}\:{dx} \\ $$

Question Number 77886 Answers: 0 Comments: 1

calculate ∫_0 ^∞ ((arctan(x^2 +x^(−2) ))/(x^2 +a^2 ))dx with a>0 2) find the value of ∫_0 ^∞ ((arctan(x^2 +x^(−2) ))/(x^2 +1))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left({x}^{\mathrm{2}} \:+{x}^{−\mathrm{2}} \right)}{{x}^{\mathrm{2}} \:+{a}^{\mathrm{2}} }{dx}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left({x}^{\mathrm{2}} \:+{x}^{−\mathrm{2}} \right)}{{x}^{\mathrm{2}} \:+\mathrm{1}}{dx} \\ $$

Question Number 77879 Answers: 0 Comments: 0

∫_1 ^∞ (1/(√(x^3 +5))) dx

$$\int_{\mathrm{1}} ^{\infty} \frac{\mathrm{1}}{\sqrt{{x}^{\mathrm{3}} +\mathrm{5}}}\:{dx} \\ $$

Question Number 77817 Answers: 0 Comments: 3

∫(dx/(x^3 (x^2 +2x+5)^4 ))

$$\int\frac{{dx}}{{x}^{\mathrm{3}} \left({x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{5}\right)^{\mathrm{4}} } \\ $$$$ \\ $$

Question Number 77816 Answers: 0 Comments: 0

∫sin(x^2 ) sin(x) dx

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

Question Number 77790 Answers: 1 Comments: 0

∫_0 ^1 (1/(√(−log(x)))) dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\sqrt{−{log}\left({x}\right)}}\:{dx} \\ $$

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