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IntegrationQuestion and Answers: Page 167

Question Number 98744 Answers: 0 Comments: 2

∫_0 ^π ∫_0 ^(2sinθ) (1+rsinθ)r dr dθ

$$\int_{\mathrm{0}} ^{\pi} \int_{\mathrm{0}} ^{\mathrm{2}{sin}\theta} \left(\mathrm{1}+{rsin}\theta\right){r}\:{dr}\:{d}\theta \\ $$

Question Number 98722 Answers: 3 Comments: 0

let f(x) =arctan((3/x)) 1) calculste f^((n)) (x) and f^((n)) (1) 2) developp f at integr seri at point x_0 =1

$$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)\:=\mathrm{arctan}\left(\frac{\mathrm{3}}{\mathrm{x}}\right) \\ $$$$\left.\mathrm{1}\right)\:\mathrm{calculste}\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{x}\right)\:\mathrm{and}\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{1}\right) \\ $$$$\left.\mathrm{2}\right)\:\mathrm{developp}\:\mathrm{f}\:\mathrm{at}\:\mathrm{integr}\:\mathrm{seri}\:\mathrm{at}\:\mathrm{point}\:\mathrm{x}_{\mathrm{0}} =\mathrm{1} \\ $$

Question Number 98721 Answers: 2 Comments: 0

calculate ∫_0 ^∞ (dx/(x^4 +x^2 +1)) 1) by using residue theorem 2) by using complex decomposition

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{\mathrm{dx}}{\mathrm{x}^{\mathrm{4}} +\mathrm{x}^{\mathrm{2}} \:+\mathrm{1}} \\ $$$$\left.\mathrm{1}\right)\:\mathrm{by}\:\mathrm{using}\:\mathrm{residue}\:\mathrm{theorem} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{by}\:\mathrm{using}\:\mathrm{complex}\:\mathrm{decomposition} \\ $$

Question Number 98713 Answers: 2 Comments: 1

∫((sin(x))/x)dx

$$\int\frac{{sin}\left({x}\right)}{{x}}{dx} \\ $$$$ \\ $$

Question Number 98679 Answers: 1 Comments: 2

prove that ∫_0 ^∞ ((3+2(√x))/(x^2 +2x+5))dx=4.13049

$${prove}\:{that}\: \\ $$$$\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{3}+\mathrm{2}\sqrt{{x}}}{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{5}}{dx}=\mathrm{4}.\mathrm{13049}\: \\ $$

Question Number 98672 Answers: 1 Comments: 0

∫_0 ^4 ∫_0 ^(x/4) e^x^2 dx dy

$$\int_{\mathrm{0}} ^{\mathrm{4}} \int_{\mathrm{0}} ^{\frac{{x}}{\mathrm{4}}} {e}^{{x}^{\mathrm{2}} } \:{dx}\:{dy} \\ $$

Question Number 98623 Answers: 3 Comments: 0

evaluate ∫_(2/(√3)) ^2 (1/(x^2 (√(4+x^2 ))))dx using the substitution x=2tanθ

$${evaluate}\: \\ $$$$\int_{\frac{\mathrm{2}}{\sqrt{\mathrm{3}}}} ^{\mathrm{2}} \frac{\mathrm{1}}{{x}^{\mathrm{2}} \sqrt{\mathrm{4}+{x}^{\mathrm{2}} }}{dx}\:{using}\:{the}\:{substitution}\:{x}=\mathrm{2tan}\theta \\ $$$$ \\ $$

Question Number 98594 Answers: 1 Comments: 0

Question Number 98589 Answers: 1 Comments: 0

calculate ∫_0 ^∞ ((sin(αx^2 ))/(x^2 +4))dx with α real

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{sin}\left(\alpha\mathrm{x}^{\mathrm{2}} \right)}{\mathrm{x}^{\mathrm{2}} \:+\mathrm{4}}\mathrm{dx}\:\:\mathrm{with}\:\alpha\:\mathrm{real} \\ $$

Question Number 98588 Answers: 0 Comments: 0

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

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

Question Number 98587 Answers: 1 Comments: 0

calculate ∫_(−∞) ^(+∞) ((cos(αx))/(x^4 +1))dx (α real)

$$\mathrm{calculate}\:\:\int_{−\infty} ^{+\infty} \:\frac{\mathrm{cos}\left(\alpha\mathrm{x}\right)}{\mathrm{x}^{\mathrm{4}} \:+\mathrm{1}}\mathrm{dx}\:\:\left(\alpha\:\mathrm{real}\right) \\ $$

Question Number 98537 Answers: 1 Comments: 1

Question Number 98531 Answers: 0 Comments: 0

Question Number 98520 Answers: 1 Comments: 0

Integrate the function f(x,y) = xy(x^2 +y^2 ) over the domain R:{−3≤x^2 −y^2 ≤3, 1≤xy≤4}

$$\mathrm{Integrate}\:\mathrm{the}\:\mathrm{function}\:\mathrm{f}\left(\mathrm{x},\mathrm{y}\right)\:=\:\mathrm{xy}\left(\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} \right) \\ $$$$\mathrm{over}\:\mathrm{the}\:\mathrm{domain}\:\mathrm{R}:\left\{−\mathrm{3}\leqslant\mathrm{x}^{\mathrm{2}} −\mathrm{y}^{\mathrm{2}} \leqslant\mathrm{3},\:\mathrm{1}\leqslant\mathrm{xy}\leqslant\mathrm{4}\right\} \\ $$

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

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

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