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IntegrationQuestion and Answers: Page 150

Question Number 101018    Answers: 0   Comments: 0

∫_(−∞) ^∞ ((log(sin^2 x))/(1+x+e^x ))dx

$$\int_{−\infty} ^{\infty} \frac{{log}\left({sin}^{\mathrm{2}} {x}\right)}{\mathrm{1}+{x}+{e}^{{x}} }{dx} \\ $$

Question Number 100969    Answers: 1   Comments: 0

find ∫_(−∞) ^∞ ((sin(cosx))/((x^2 −x+1)^2 ))dx

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

Question Number 100967    Answers: 1   Comments: 0

calculate ∫_0 ^(π/2) ln(2+ sinθ)dθ

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{ln}\left(\mathrm{2}+\:\mathrm{sin}\theta\right)\mathrm{d}\theta \\ $$

Question Number 100965    Answers: 0   Comments: 0

calculate ∫_0 ^π ln(x^2 −2xcosθ +1)dθ (x real)

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

Question Number 100956    Answers: 2   Comments: 0

Question Number 100948    Answers: 0   Comments: 1

Question Number 100829    Answers: 0   Comments: 0

hello every one prove that ∫_0 ^(π/2) cos^u (x) cos(ax) arctan(b cos(x)) dx =((2^(−u−2) .π.b.Γ(u+2))/(Γ(((u−a+3)/2))Γ(((u+a+3)/2)))).x_4 F_3 ((((1/2),1+(u/2),((u+3)/2),−b^2 )),(((3/2),((u−a+3)/2),((u+a+3)/2))) ) Re u>−1 ,∣arg(1+b^2 ) ∣<π

$${hello}\:{every}\:{one}\: \\ $$$$ \\ $$$${prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {cos}^{{u}} \left({x}\right)\:{cos}\left({ax}\right)\:{arctan}\left({b}\:{cos}\left({x}\right)\right)\:{dx} \\ $$$$=\frac{\mathrm{2}^{−{u}−\mathrm{2}} .\pi.{b}.\Gamma\left({u}+\mathrm{2}\right)}{\Gamma\left(\frac{{u}−{a}+\mathrm{3}}{\mathrm{2}}\right)\Gamma\left(\frac{{u}+{a}+\mathrm{3}}{\mathrm{2}}\right)}.{x}_{\mathrm{4}} {F}_{\mathrm{3}} \begin{pmatrix}{\frac{\mathrm{1}}{\mathrm{2}},\mathrm{1}+\frac{{u}}{\mathrm{2}},\frac{{u}+\mathrm{3}}{\mathrm{2}},−{b}^{\mathrm{2}} }\\{\frac{\mathrm{3}}{\mathrm{2}},\frac{{u}−{a}+\mathrm{3}}{\mathrm{2}},\frac{{u}+{a}+\mathrm{3}}{\mathrm{2}}}\end{pmatrix} \\ $$$$ \\ $$$$ \\ $$$${Re}\:{u}>−\mathrm{1}\:,\mid{arg}\left(\mathrm{1}+{b}^{\mathrm{2}} \right)\:\mid<\pi \\ $$$$ \\ $$

Question Number 100789    Answers: 2   Comments: 0

Question Number 100746    Answers: 1   Comments: 0

∫_(−∞) ^∞ ((cos3x)/((1+x^2 )^2 ))dx

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

Question Number 100657    Answers: 1   Comments: 3

∫ ((3x−1)/(x^2 +9)) dx

$$\int\:\:\frac{\mathrm{3}{x}−\mathrm{1}}{{x}^{\mathrm{2}} +\mathrm{9}}\:{dx} \\ $$

Question Number 100653    Answers: 1   Comments: 0

Question Number 100613    Answers: 0   Comments: 0

Question Number 100606    Answers: 0   Comments: 0

∫e^(ix^(ix...∞) ) dx

$$\int{e}^{{ix}^{{ix}...\infty} } {dx} \\ $$

Question Number 100590    Answers: 2   Comments: 0

∫_0 ^∞ (dx/((1+x^(18) )^2 ))

$$\int_{\mathrm{0}} ^{\infty} \:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{18}} \right)^{\mathrm{2}} } \\ $$

Question Number 100584    Answers: 1   Comments: 0

∫i^i^(i......∞) dx

$$\int{i}^{{i}^{{i}......\infty} } {dx} \\ $$

Question Number 100557    Answers: 2   Comments: 0

Ω=∫_0 ^∞ (e^(ax) /(e^(bx) +1))dx, b>a

$$\Omega=\underset{\mathrm{0}} {\overset{\infty} {\int}}\:\frac{{e}^{{ax}} }{{e}^{{bx}} +\mathrm{1}}{dx},\:{b}>{a} \\ $$

Question Number 100538    Answers: 0   Comments: 1

Question Number 100543    Answers: 2   Comments: 1

Question Number 100514    Answers: 2   Comments: 0

calculatelim_(n→+∞) ∫_0 ^∞ (1−(x/n))^n ln(1+2x)dx

$$\mathrm{calculatelim}_{\mathrm{n}\rightarrow+\infty} \:\int_{\mathrm{0}} ^{\infty} \:\left(\mathrm{1}−\frac{\mathrm{x}}{\mathrm{n}}\right)^{\mathrm{n}} \mathrm{ln}\left(\mathrm{1}+\mathrm{2x}\right)\mathrm{dx} \\ $$

Question Number 100513    Answers: 0   Comments: 0

findA_(nm) =∫_0 ^∞ e^(−nx) ∣sin(px)∣ dx with n and p integr natural ≥1

$$\mathrm{findA}_{\mathrm{nm}} \:=\int_{\mathrm{0}} ^{\infty} \:\:\mathrm{e}^{−\mathrm{nx}} \:\mid\mathrm{sin}\left(\mathrm{px}\right)\mid\:\mathrm{dx}\:\:\mathrm{with}\:\:\mathrm{n}\:\mathrm{and}\:\mathrm{p}\:\mathrm{integr}\:\mathrm{natural}\:\geqslant\mathrm{1} \\ $$

Question Number 100512    Answers: 0   Comments: 0

calculate ∫_(−∞) ^(+∞) (x^n /((x^2 +x+1)^n )) dx with n integr and n≥2

$$\mathrm{calculate}\:\int_{−\infty} ^{+\infty} \:\frac{\mathrm{x}^{\mathrm{n}} }{\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{x}+\mathrm{1}\right)^{\mathrm{n}} }\:\mathrm{dx}\:\:\mathrm{with}\:\mathrm{n}\:\mathrm{integr}\:\mathrm{and}\:\mathrm{n}\geqslant\mathrm{2} \\ $$

Question Number 100511    Answers: 1   Comments: 0

calculate ∫_(−∞) ^∞ ((arctan(cosx +sinx))/(x^2 +4)) dx

$$\mathrm{calculate}\:\:\int_{−\infty} ^{\infty} \:\:\frac{\mathrm{arctan}\left(\mathrm{cosx}\:+\mathrm{sinx}\right)}{\mathrm{x}^{\mathrm{2}} \:+\mathrm{4}}\:\mathrm{dx} \\ $$

Question Number 100468    Answers: 3   Comments: 0

Σ_(n=1) ^∞ (n/((2n+1)!)) help me pls

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{n}}{\left(\mathrm{2}{n}+\mathrm{1}\right)!} \\ $$$${help}\:{me}\:{pls} \\ $$

Question Number 100450    Answers: 1   Comments: 0

Question Number 100438    Answers: 0   Comments: 5

∫_0 ^1 ∫_0 ^x x^2 y^(xy) dydx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{{x}} {x}^{\mathrm{2}} {y}^{{xy}} {dydx} \\ $$

Question Number 100362    Answers: 3   Comments: 0

∫_0 ^1 ∫_0 ^1 e^(2x+y) dydx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} {e}^{\mathrm{2}{x}+{y}} {dydx} \\ $$

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