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IntegrationQuestion and Answers: Page 131

Question Number 118010 Answers: 3 Comments: 1

∫(dx/(x^4 +x^2 +1))

$$\int\frac{{dx}}{{x}^{\mathrm{4}} +{x}^{\mathrm{2}} +\mathrm{1}} \\ $$

Question Number 117979 Answers: 1 Comments: 1

... nice calculus... prove that: ∣ Γ ( i ) ∣=^? (√(π/(sinh(π)))) Γ: Euler gamma function ...m.n.july.1970...

$$\:\:\:\:\:\:\:\:\:\:...\:{nice}\:\:{calculus}... \\ $$$$\:\:\:\:\:{prove}\:\:{that}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\mid\:\:\Gamma\:\left(\:{i}\:\right)\:\mid\overset{?} {=}\:\sqrt{\frac{\pi}{{sinh}\left(\pi\right)}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\Gamma:\:\mathscr{E}{uler}\:{gamma}\:{function}\:\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{m}.{n}.{july}.\mathrm{1970}... \\ $$

Question Number 117963 Answers: 2 Comments: 0

Question Number 117948 Answers: 4 Comments: 0

... nice integral... please evaluate :: I =∫_0 ^( 1) (sin(x)+sin((1/x)))(dx/x) =?? m.n.1970

$$\:\:\:\:\:\:\:\:...\:\:{nice}\:\:{integral}...\: \\ $$$$\:\:\:{please}\:{evaluate}\::: \\ $$$$ \\ $$$$\:\:\:\:\mathrm{I}\:=\int_{\mathrm{0}} ^{\:\mathrm{1}} \left({sin}\left({x}\right)+{sin}\left(\frac{\mathrm{1}}{{x}}\right)\right)\frac{{dx}}{{x}}\:=?? \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:{m}.{n}.\mathrm{1970} \\ $$$$\:\: \\ $$

Question Number 117945 Answers: 3 Comments: 1

f(x) = ∫ ((5x^8 +7x^6 )/((2x^7 +x^2 +1)^2 )) dx and f(0) = 0 , then f(1) = _

$$\mathrm{f}\left(\mathrm{x}\right)\:=\:\int\:\frac{\mathrm{5x}^{\mathrm{8}} +\mathrm{7x}^{\mathrm{6}} }{\left(\mathrm{2x}^{\mathrm{7}} +\mathrm{x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} }\:\mathrm{dx}\:\mathrm{and}\:\: \\ $$$$\mathrm{f}\left(\mathrm{0}\right)\:=\:\mathrm{0}\:,\:\mathrm{then}\:\mathrm{f}\left(\mathrm{1}\right)\:=\:\_\: \\ $$

Question Number 117944 Answers: 3 Comments: 0

Find the value of k satisfies the equation ∫ _0^(π/3) (((tan x (√(cos x)))/( (√(2k)))) ) dx = 1−(1/( (√2)))

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{k}\:\mathrm{satisfies}\: \\ $$$$\mathrm{the}\:\mathrm{equation}\:\int\:_{\mathrm{0}} ^{\frac{\pi}{\mathrm{3}}} \:\left(\frac{\mathrm{tan}\:\mathrm{x}\:\sqrt{\mathrm{cos}\:\mathrm{x}}}{\:\sqrt{\mathrm{2k}}}\:\right)\:\mathrm{dx}\:=\:\mathrm{1}−\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}} \\ $$$$ \\ $$

Question Number 117910 Answers: 3 Comments: 1

∫ sin^(−1) ((√x)) dx =?

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

Question Number 117841 Answers: 1 Comments: 0

∫ ln (1−e^(−2x) ) dx =?

$$\int\:\mathrm{ln}\:\left(\mathrm{1}−{e}^{−\mathrm{2}{x}} \right)\:{dx}\:=? \\ $$

Question Number 117817 Answers: 2 Comments: 1

Question Number 117811 Answers: 1 Comments: 0

∫_( 0) ^( 𝛑) ln∣sinh(x)∣dx

$$\int_{\:\mathrm{0}} ^{\:\boldsymbol{\pi}} \boldsymbol{\mathrm{ln}}\mid\boldsymbol{\mathrm{sinh}}\left(\boldsymbol{\mathrm{x}}\right)\mid\boldsymbol{\mathrm{dx}} \\ $$

Question Number 117806 Answers: 4 Comments: 2

... nice calculus... i :: 1 +(4/9)+(9/(36))+((16)/(100))+...= ?? ii:: ∫_0 ^( (π/2)) x^2 cot(x) dx=?? m.n.1970

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\:{nice}\:\:{calculus}... \\ $$$$\:\:\:\:{i}\:::\:\:\:\mathrm{1}\:+\frac{\mathrm{4}}{\mathrm{9}}+\frac{\mathrm{9}}{\mathrm{36}}+\frac{\mathrm{16}}{\mathrm{100}}+...=\:?? \\ $$$$\:\:\:\:\:{ii}::\:\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} {x}^{\mathrm{2}} {cot}\left({x}\right)\:{dx}=?? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{m}.{n}.\mathrm{1970} \\ $$$$ \\ $$

Question Number 117724 Answers: 2 Comments: 5

∫ ((sin^(−1) (x))/x^2 ) dx =?

$$\int\:\frac{\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{x}\right)}{\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx}\:=? \\ $$

Question Number 117655 Answers: 0 Comments: 3

Question Number 117654 Answers: 0 Comments: 1

Question Number 117638 Answers: 2 Comments: 0

∫_0 ^1 ((2x^(12) +5x^9 )/((x^5 +x^3 +1)^3 )) dx =?

$$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\frac{\mathrm{2x}^{\mathrm{12}} +\mathrm{5x}^{\mathrm{9}} }{\left(\mathrm{x}^{\mathrm{5}} +\mathrm{x}^{\mathrm{3}} +\mathrm{1}\right)^{\mathrm{3}} }\:\mathrm{dx}\:=?\: \\ $$

Question Number 117574 Answers: 1 Comments: 0

... advanced integral... Evaluate :: I := ∫_0 ^( ∞) (( 4xln(x))/(x^4 +2x^2 +4 ))dx =?? ... m.n.1970..

$$\:\:\:\:\:\:\:\:...\:{advanced}\:\:{integral}... \\ $$$$\:\:\:\:\:\: \\ $$$$\mathscr{E}{valuate}\:::\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{I}\::=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:\mathrm{4}{xln}\left({x}\right)}{{x}^{\mathrm{4}} +\mathrm{2}{x}^{\mathrm{2}} +\mathrm{4}\:}{dx}\:=??\: \\ $$$$\:\:\:\:\:...\:{m}.{n}.\mathrm{1970}.. \\ $$$$\: \\ $$

Question Number 117527 Answers: 1 Comments: 0

∫_( 0) ^( 1) xsec(2x)dx

$$\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \mathrm{xsec}\left(\mathrm{2x}\right)\mathrm{dx} \\ $$

Question Number 117511 Answers: 3 Comments: 0

Question Number 117496 Answers: 2 Comments: 0

∫ ((sec^2 θ tan^2 θ)/( (√(9−tan^2 θ)))) dθ =?

$$\int\:\frac{\mathrm{sec}\:^{\mathrm{2}} \theta\:\mathrm{tan}\:^{\mathrm{2}} \theta}{\:\sqrt{\mathrm{9}−\mathrm{tan}\:^{\mathrm{2}} \theta}}\:\mathrm{d}\theta\:=? \\ $$

Question Number 117463 Answers: 0 Comments: 2

Question Number 117446 Answers: 2 Comments: 0

Evaluate ∫((3x^2 −5)/(x^4 +6x^2 +25))dx

$$\mathrm{Evaluate}\:\int\frac{\mathrm{3}{x}^{\mathrm{2}} −\mathrm{5}}{{x}^{\mathrm{4}} +\mathrm{6}{x}^{\mathrm{2}} +\mathrm{25}}\mathrm{d}{x} \\ $$

Question Number 117437 Answers: 2 Comments: 0

∫_0 ^∞ (dx/(a^3 +x^3 )) generaly ∫_0 ^∞ (dx/(p+x^n ))

$$\int_{\mathrm{0}} ^{\infty} \frac{{dx}}{{a}^{\mathrm{3}} +{x}^{\mathrm{3}} } \\ $$$$ \\ $$$${generaly} \\ $$$$\int_{\mathrm{0}} ^{\infty} \frac{{dx}}{{p}+{x}^{{n}} } \\ $$

Question Number 117431 Answers: 0 Comments: 0

Question Number 117409 Answers: 1 Comments: 0

Question Number 117403 Answers: 1 Comments: 1

∫_0 ^1 (arc tan x)^2 dx =?

$$\:\:\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\left(\mathrm{arc}\:\mathrm{tan}\:\mathrm{x}\right)^{\mathrm{2}} \:\mathrm{dx}\:=? \\ $$

Question Number 117396 Answers: 3 Comments: 0

...differential equation... solve : (dy/dx)=(1/(xy+2x^2 y)) general solution =??? m.n.1970

$$\:\:\:\:\:\:\:\:...{differential}\:\:{equation}...\: \\ $$$$ \\ $$$$\:\:\:\:{solve}\:: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\frac{{dy}}{{dx}}=\frac{\mathrm{1}}{{xy}+\mathrm{2}{x}^{\mathrm{2}} {y}} \\ $$$$\:\:\:\:\:\:\:\:\:{general}\:\:{solution}\:=??? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{m}.{n}.\mathrm{1970} \\ $$$$\: \\ $$

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