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IntegrationQuestion and Answers: Page 194

Question Number 79825 Answers: 0 Comments: 4

∫_( 0) ^( 1) (√(x^3 + 1)) dx

$$\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\sqrt{\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{1}}\:\:\mathrm{dx} \\ $$

Question Number 79824 Answers: 2 Comments: 7

Question Number 79814 Answers: 0 Comments: 5

Question Number 79763 Answers: 1 Comments: 2

calculate ∫_0 ^π {cos^8 x +sin^8 x}dx

$${calculate}\:\int_{\mathrm{0}} ^{\pi} \left\{{cos}^{\mathrm{8}} {x}\:+{sin}^{\mathrm{8}} {x}\right\}{dx} \\ $$

Question Number 79758 Answers: 0 Comments: 1

find value of ∫_0 ^1 ln(1+ix^2 )dx and ∫_0 ^1 ln(1−ix^2 )dx with i=(√(−1))

$${find}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+{ix}^{\mathrm{2}} \right){dx}\:{and}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}−{ix}^{\mathrm{2}} \right){dx}\:{with}\:{i}=\sqrt{−\mathrm{1}} \\ $$

Question Number 79730 Answers: 1 Comments: 1

I) For witch value of α the integral C=∫_0 ^( ∞) ((1/(√(1+2x^2 )))−(1/(x+1)))dx conveege ? And in this case calculate α. II) Let Δ={(x; y)/ ∣x∣+∣y∣≤2} a) Calculate I_1 = ∫∫_Δ dxdy and ∫∫_Δ ((dxdy)/((∣x∣+∣y∣)^2 +4))

$$\left.{I}\right)\:\:{For}\:{witch}\:{value}\:{of}\:\alpha\:{the}\:{integral} \\ $$$$\:{C}=\int_{\mathrm{0}} ^{\:\infty} \left(\frac{\mathrm{1}}{\sqrt{\mathrm{1}+\mathrm{2}{x}^{\mathrm{2}} }}−\frac{\mathrm{1}}{{x}+\mathrm{1}}\right){dx}\:\:{conveege}\:\:? \\ $$$${And}\:{in}\:{this}\:{case}\:{calculate}\:\alpha. \\ $$$$\left.{II}\right)\:\:{Let}\:\Delta=\left\{\left({x};\:{y}\right)/\:\mid{x}\mid+\mid{y}\mid\leqslant\mathrm{2}\right\} \\ $$$$\left.\:\:\:\:\:{a}\right)\:{Calculate}\:{I}_{\mathrm{1}} =\:\int\int_{\Delta} {dxdy}\:\:\:{and}\:\:\int\int_{\Delta} \frac{{dxdy}}{\left(\mid{x}\mid+\mid{y}\mid\right)^{\mathrm{2}} +\mathrm{4}} \\ $$

Question Number 79646 Answers: 0 Comments: 1

calculate A_n =∫_0 ^1 cos(narcosx)dx with n integr natural

$${calculate}\:\:{A}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:{cos}\left({narcosx}\right){dx} \\ $$$${with}\:{n}\:{integr}\:{natural} \\ $$

Question Number 79645 Answers: 0 Comments: 0

find ∫_0 ^1 ln(1+x^4 )dx

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{ln}\left(\mathrm{1}+{x}^{\mathrm{4}} \right){dx} \\ $$

Question Number 79634 Answers: 1 Comments: 3

Question Number 79627 Answers: 0 Comments: 2

1) expicite f(x)=∫_0 ^1 ((ln(1+xt^2 ))/(1+t^2 ))dt with x≥0 2)calculate ∫_0 ^1 ((ln(1+t^2 ))/(1+t^2 ))dt and ∫_0 ^1 ((ln(1+2t^2 ))/(1+t^2 ))dt

$$\left.\mathrm{1}\right)\:{expicite}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}+{xt}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}\:{with}\:{x}\geqslant\mathrm{0} \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}\:{and}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}+\mathrm{2}{t}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$

Question Number 79615 Answers: 0 Comments: 3

prove that with using hypergeometric function ∫_0 ^π sin(x^2 )=(π^3 /3) 1F_2 [(3/4);(3/2);(7/4);((−π^4 )/4)]

$${prove}\:{that}\:{with}\:{using}\:{hypergeometric}\:{function} \\ $$$$\int_{\mathrm{0}} ^{\pi} {sin}\left({x}^{\mathrm{2}} \right)=\frac{\pi^{\mathrm{3}} }{\mathrm{3}}\:\mathrm{1}{F}_{\mathrm{2}} \left[\frac{\mathrm{3}}{\mathrm{4}};\frac{\mathrm{3}}{\mathrm{2}};\frac{\mathrm{7}}{\mathrm{4}};\frac{−\pi^{\mathrm{4}} }{\mathrm{4}}\right]\: \\ $$

Question Number 79607 Answers: 0 Comments: 1

Solve this ∫_ (((x−yz))/((x^2 +y^2 −2xyz)^(3/2) ))dz

$${Solve}\:{this} \\ $$$$\int_{} \frac{\left({x}−{yz}\right)}{\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} −\mathrm{2}{xyz}\right)^{\mathrm{3}/\mathrm{2}} }{dz} \\ $$$$ \\ $$$$ \\ $$

Question Number 79580 Answers: 0 Comments: 5

does this matter reasonable ∫ sin^x (x) dx ?

$$\mathrm{does}\:\mathrm{this}\:\mathrm{matter}\:\mathrm{reasonable} \\ $$$$\int\:\mathrm{sin}\:^{\mathrm{x}} \left(\mathrm{x}\right)\:\mathrm{dx}\:? \\ $$

Question Number 79612 Answers: 1 Comments: 0

∫ (dx/((√(x ))((x)^(1/(4 )) +1)^(10) )) = ?

$$\int\:\frac{\mathrm{dx}}{\sqrt{\mathrm{x}\:}\left(\sqrt[{\mathrm{4}\:}]{\mathrm{x}}+\mathrm{1}\right)^{\mathrm{10}} }\:=\:? \\ $$

Question Number 79531 Answers: 0 Comments: 1

∫^1 _0 ((ln((1/x)+x))/(x^2 +1))dx ?

$$\underset{\mathrm{0}} {\int}^{\mathrm{1}} \:\frac{\mathrm{ln}\left(\frac{\mathrm{1}}{\mathrm{x}}+\mathrm{x}\right)}{\mathrm{x}^{\mathrm{2}} +\mathrm{1}}\mathrm{dx}\:? \\ $$

Question Number 79528 Answers: 0 Comments: 0

find ∫_0 ^∞ e^(−x^3 ) cos(x^2 )dx

$${find}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{x}^{\mathrm{3}} } {cos}\left({x}^{\mathrm{2}} \right){dx} \\ $$

Question Number 79527 Answers: 0 Comments: 1

find ∫_0 ^∞ e^(−x^3 ) dx

$${find}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{x}^{\mathrm{3}} } {dx} \\ $$

Question Number 79520 Answers: 1 Comments: 2

Question Number 79516 Answers: 0 Comments: 3

Question Number 79500 Answers: 1 Comments: 0

∫(cot^2 x+cot^4 x)dx

$$\int\left(\mathrm{cot}\:^{\mathrm{2}} {x}+\mathrm{cot}\:^{\mathrm{4}} {x}\right){dx} \\ $$

Question Number 79485 Answers: 1 Comments: 0

∫(tan^2 x+tan^4 x)dx

$$\int\left(\mathrm{tan}\:^{\mathrm{2}} {x}+\mathrm{tan}\:^{\mathrm{4}} {x}\right){dx} \\ $$

Question Number 79413 Answers: 0 Comments: 1

Question Number 79373 Answers: 0 Comments: 0

find ∫_0 ^∞ e^(−(x^3 +(1/x^3 ))) dx

$${find}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−\left({x}^{\mathrm{3}} \:+\frac{\mathrm{1}}{{x}^{\mathrm{3}} }\right)} {dx} \\ $$

Question Number 79352 Answers: 1 Comments: 2

∫(dx/(1−(√(cos(x)))))

$$\int\frac{{dx}}{\mathrm{1}−\sqrt{{cos}\left({x}\right)}} \\ $$

Question Number 79325 Answers: 0 Comments: 2

Convergence of : 1) I=∫_1 ^( ∞) ((e^(−t/5) ∣sin(lnt)∣)/((t−1)^(3/2) ))dt 2) I=∫_1 ^∞ ((√(lnx))/((x−1)(√x)))dx

$$\:\boldsymbol{{Convergence}}\:\:\boldsymbol{{of}}\:: \\ $$$$\left.\:\:\mathrm{1}\right)\:\:\:\boldsymbol{{I}}=\int_{\mathrm{1}} ^{\:\infty} \frac{\boldsymbol{{e}}^{−\boldsymbol{{t}}/\mathrm{5}} \mid\boldsymbol{{sin}}\left(\boldsymbol{{lnt}}\right)\mid}{\left(\boldsymbol{{t}}−\mathrm{1}\right)^{\mathrm{3}/\mathrm{2}} }\boldsymbol{{dt}} \\ $$$$\left.\:\:\mathrm{2}\right)\:\:\:\boldsymbol{{I}}=\int_{\mathrm{1}} ^{\infty} \frac{\sqrt{\boldsymbol{{lnx}}}}{\left(\boldsymbol{{x}}−\mathrm{1}\right)\sqrt{\boldsymbol{{x}}}}\boldsymbol{{dx}} \\ $$

Question Number 79222 Answers: 0 Comments: 3

∫_0 ^1 (x^n /(Σ_(k=0) ^(n−1) x^k ))dx=?

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{{n}} }{\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\sum}}{x}^{{k}} }{dx}=? \\ $$

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