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IntegrationQuestion and Answers: Page 166

Question Number 98463 Answers: 1 Comments: 0

Question Number 98445 Answers: 2 Comments: 0

give at form of serie U_n =∫_0 ^1 ((x^n ln(x))/((1+x)^2 ))dx

$$\mathrm{give}\:\mathrm{at}\:\mathrm{form}\:\mathrm{of}\:\mathrm{serie}\:\mathrm{U}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\mathrm{x}^{\mathrm{n}} \mathrm{ln}\left(\mathrm{x}\right)}{\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{2}} }\mathrm{dx} \\ $$

Question Number 98444 Answers: 1 Comments: 0

calculate ∫_0 ^1 ((lnx)/((1+x)^2 ))dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\mathrm{lnx}}{\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{2}} }\mathrm{dx} \\ $$

Question Number 98428 Answers: 1 Comments: 0

let f(x)=x^2 ,2π periodi even developp f at fourier serie

$$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}^{\mathrm{2}} \:\:,\mathrm{2}\pi\:\mathrm{periodi}\:\mathrm{even}\:\mathrm{developp}\:\mathrm{f}\:\mathrm{at}\:\mathrm{fourier}\:\mathrm{serie} \\ $$

Question Number 98426 Answers: 2 Comments: 0

let f(x) =∫_(π/4) ^(π/3) (dt/(x+tant)) calculate f(x) 2)explicit g(x) =∫_(π/4) ^(π/3) (dt/((x+tant)^2 )) 3) find the value of integrals ∫_(π/4) ^(π/3) (dt/(2+tant)) and ∫_(π/4) ^(π/3) (dt/((2+tant)^2 ))

$$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)\:=\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \:\frac{\mathrm{dt}}{\mathrm{x}+\mathrm{tant}}\:\:\mathrm{calculate}\:\mathrm{f}\left(\mathrm{x}\right) \\ $$$$\left.\mathrm{2}\right)\mathrm{explicit}\:\mathrm{g}\left(\mathrm{x}\right)\:=\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \:\frac{\mathrm{dt}}{\left(\mathrm{x}+\mathrm{tant}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right)\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{integrals}\:\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \:\frac{\mathrm{dt}}{\mathrm{2}+\mathrm{tant}}\:\mathrm{and}\:\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \:\frac{\mathrm{dt}}{\left(\mathrm{2}+\mathrm{tant}\right)^{\mathrm{2}} } \\ $$

Question Number 98423 Answers: 1 Comments: 0

calculate ∫_0 ^∞ e^(−x^2 −(1/x^2 )) dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\mathrm{e}^{−\mathrm{x}^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }} \mathrm{dx} \\ $$

Question Number 98382 Answers: 0 Comments: 2

∫ tan x (√(1+tan^4 x)) dx

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

Question Number 98338 Answers: 2 Comments: 0

∫cos(x^(18) ) dx

$$\int{cos}\left({x}^{\mathrm{18}} \right)\:{dx} \\ $$$$ \\ $$

Question Number 98311 Answers: 1 Comments: 0

calculate ∫_0 ^∞ ((lnx)/((1+x)^2 ))dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{lnx}}{\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{2}} }\mathrm{dx}\: \\ $$

Question Number 98305 Answers: 2 Comments: 0

calculate ∫_0 ^∞ ((ln(x))/((1+x)^3 ))dx

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

Question Number 98271 Answers: 2 Comments: 5

GivenU_n =∫_0 ^1 x^n (√(1−x))dx n∈N, show that U_n =((2^(n+2) n!(n+1))/((2n+3)!))

$$\mathcal{G}\mathrm{ivenU}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{x}^{\mathrm{n}} \sqrt{\mathrm{1}−\mathrm{x}}\mathrm{dx}\:\:\mathrm{n}\in\mathbb{N},\:\mathrm{show}\:\mathrm{that} \\ $$$$\mathrm{U}_{\mathrm{n}} =\frac{\mathrm{2}^{\mathrm{n}+\mathrm{2}} \mathrm{n}!\left(\mathrm{n}+\mathrm{1}\right)}{\left(\mathrm{2n}+\mathrm{3}\right)!} \\ $$

Question Number 98256 Answers: 2 Comments: 0

∫_0 ^∞ ((log(x))/((√x)(x+1)^2 ))dx

$$\int_{\mathrm{0}} ^{\infty} \frac{{log}\left({x}\right)}{\sqrt{{x}}\left({x}+\mathrm{1}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 98249 Answers: 0 Comments: 0

explicit A(θ) =∫_1 ^(+∞) ((ln(lnx))/(x^2 +2xcosθ +1))dx with −π<θ<π

$$\mathrm{explicit}\:\mathrm{A}\left(\theta\right)\:=\int_{\mathrm{1}} ^{+\infty} \:\frac{\mathrm{ln}\left(\mathrm{lnx}\right)}{\mathrm{x}^{\mathrm{2}} \:+\mathrm{2xcos}\theta\:+\mathrm{1}}\mathrm{dx}\:\:\:\mathrm{with}\:−\pi<\theta<\pi \\ $$

Question Number 98248 Answers: 0 Comments: 0

find the value of ∫_1 ^(+∞) ((ln(lnx))/(x^2 +1))dx

$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\int_{\mathrm{1}} ^{+\infty} \:\frac{\mathrm{ln}\left(\mathrm{lnx}\right)}{\mathrm{x}^{\mathrm{2}} \:+\mathrm{1}}\mathrm{dx} \\ $$

Question Number 98246 Answers: 1 Comments: 0

∫(x/(sin^2 (x−3)))dx

$$\int\frac{{x}}{{sin}^{\mathrm{2}} \left({x}−\mathrm{3}\right)}{dx} \\ $$

Question Number 98245 Answers: 0 Comments: 0

lim_(k→0) ∫_0 ^k (1/(√(cos(x)−cos(k))))dx=?

$$\underset{{k}\rightarrow\mathrm{0}} {{lim}}\int_{\mathrm{0}} ^{{k}} \frac{\mathrm{1}}{\sqrt{{cos}\left({x}\right)−{cos}\left({k}\right)}}{dx}=? \\ $$

Question Number 98214 Answers: 2 Comments: 2

Question Number 98182 Answers: 2 Comments: 0

find ∫ x^2 (√((2−x)/(2+x)))dx

$$\mathrm{find}\:\int\:\mathrm{x}^{\mathrm{2}} \sqrt{\frac{\mathrm{2}−\mathrm{x}}{\mathrm{2}+\mathrm{x}}}\mathrm{dx} \\ $$

Question Number 98181 Answers: 0 Comments: 0

calculate ∫_0 ^π ln(x^2 −2xcosθ +1)dθ

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\pi} \mathrm{ln}\left(\mathrm{x}^{\mathrm{2}} −\mathrm{2xcos}\theta\:+\mathrm{1}\right)\mathrm{d}\theta \\ $$

Question Number 98179 Answers: 2 Comments: 0

calculate ∫_1 ^(+∞) (dx/(x^2 (1−x^2 )^3 ))

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

Question Number 98151 Answers: 2 Comments: 0

∫e^(x^5 +8x^2 ) dx =((√π)/(4(√2)))e^x^5 erfi(2(√2)x)−((5(√π))/(4(128)(√2)))(super−erf_((hyper)) (2(√2)x))+c where[super−erf_((hyper)) (t)] is super−function in D_2 and [D_n ]

$$\int{e}^{{x}^{\mathrm{5}} +\mathrm{8}{x}^{\mathrm{2}} } {dx} \\ $$$$=\frac{\sqrt{\pi}}{\mathrm{4}\sqrt{\mathrm{2}}}{e}^{{x}^{\mathrm{5}} } {erfi}\left(\mathrm{2}\sqrt{\mathrm{2}}{x}\right)−\frac{\mathrm{5}\sqrt{\pi}}{\mathrm{4}\left(\mathrm{128}\right)\sqrt{\mathrm{2}}}\left({super}−{erf}_{\left({hyper}\right)} \left(\mathrm{2}\sqrt{\mathrm{2}}{x}\right)\right)+{c} \\ $$$$ \\ $$$${where}\left[{super}−{erf}_{\left({hyper}\right)} \left({t}\right)\right]\:{is}\:{super}−{function} \\ $$$${in}\:{D}_{\mathrm{2}} \:{and}\:\left[{D}_{{n}} \right] \\ $$

Question Number 98105 Answers: 2 Comments: 0

calculate ∫_3 ^(+∞) (((x+1)dx)/((x−2)^2 ( 2x+3)^3 ))

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

Question Number 98020 Answers: 1 Comments: 0

What is the area of the region bounded by x^2 +y^2 ≤ 9 ; x+y ≤ 3 and y ≤ x

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{the}\:\mathrm{region}\:\mathrm{bounded} \\ $$$$\mathrm{by}\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} \:\leqslant\:\mathrm{9}\:;\:\mathrm{x}+\mathrm{y}\:\leqslant\:\mathrm{3}\:\mathrm{and}\:\mathrm{y}\:\leqslant\:\mathrm{x}\: \\ $$

Question Number 98016 Answers: 1 Comments: 3

∫_0 ^1 ((ln^2 (x))/(x^2 +1)) dx ?

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

Question Number 97928 Answers: 1 Comments: 0

find the general formula ∫_0 ^(π/2) tan^α (x) dx

$${find}\:{the}\:{general}\:{formula} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {tan}^{\alpha} \left({x}\right)\:{dx} \\ $$

Question Number 97839 Answers: 3 Comments: 1

calculate A_n =∫_0 ^((nπ)/4) (dx/(3cos^4 x +3sin^4 x−1))

$$\mathrm{calculate}\:\mathrm{A}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\frac{\mathrm{n}\pi}{\mathrm{4}}} \:\frac{\mathrm{dx}}{\mathrm{3cos}^{\mathrm{4}} \mathrm{x}\:+\mathrm{3sin}^{\mathrm{4}} \mathrm{x}−\mathrm{1}} \\ $$

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