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IntegrationQuestion and Answers: Page 163

Question Number 101271    Answers: 0   Comments: 2

find ∫ ((xdx)/((√(x^2 +x+1))+(√(x^2 −x+1))))

$$\mathrm{find}\:\int\:\:\:\frac{\mathrm{xdx}}{\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}}+\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{1}}} \\ $$

Question Number 101270    Answers: 0   Comments: 0

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

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

Question Number 101269    Answers: 1   Comments: 0

calculate ∫_4 ^(+∞) (dx/((x−2)^5 (x+3)^7 ))

$$\mathrm{calculate}\:\int_{\mathrm{4}} ^{+\infty} \:\:\:\:\:\frac{\mathrm{dx}}{\left(\mathrm{x}−\mathrm{2}\right)^{\mathrm{5}} \left(\mathrm{x}+\mathrm{3}\right)^{\mathrm{7}} } \\ $$

Question Number 101268    Answers: 1   Comments: 0

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

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

Question Number 101266    Answers: 0   Comments: 0

calculate ∫_1 ^(+∞) (dx/(x^2 (x+1)^3 (x+2)^4 ))

$$\mathrm{calculate}\:\:\int_{\mathrm{1}} ^{+\infty} \:\:\:\:\:\:\frac{\mathrm{dx}}{\mathrm{x}^{\mathrm{2}} \left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{3}} \left(\mathrm{x}+\mathrm{2}\right)^{\mathrm{4}} } \\ $$

Question Number 101286    Answers: 0   Comments: 3

∫(((x^m −x^n )^2 )/(√x))dx=?

$$\int\frac{\left(\mathrm{x}^{\mathrm{m}} −\mathrm{x}^{\mathrm{n}} \right)^{\mathrm{2}} }{\sqrt{\mathrm{x}}}\mathrm{dx}=? \\ $$

Question Number 101234    Answers: 0   Comments: 0

Show that the greatest integer function is Riemann integrable within all segments of R

$$\mathcal{S}\mathrm{how}\:\mathrm{that}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{integer}\:\mathrm{function}\:\mathrm{is}\:\mathrm{Riemann} \\ $$$$\mathrm{integrable}\:\mathrm{within}\:\mathrm{all}\:\mathrm{segments}\:\mathrm{of}\:\mathbb{R} \\ $$

Question Number 101220    Answers: 1   Comments: 0

∫∫_D (√(x^2 +y^2 ))dxdy D= { (((x,y)∈R, x^2 +y^2 ≥2y, x^2 +y^2 ≤1)),((x≥0 , y≥0)) :}

$$\int\int_{\mathrm{D}} \sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} }\mathrm{dxdy}\:\:\:\mathcal{D}=\begin{cases}{\left(\mathrm{x},\mathrm{y}\right)\in\mathbb{R},\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} \geqslant\mathrm{2y},\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} \leqslant\mathrm{1}}\\{\mathrm{x}\geqslant\mathrm{0}\:,\:\mathrm{y}\geqslant\mathrm{0}}\end{cases} \\ $$

Question Number 101285    Answers: 0   Comments: 1

∫(((x^m −x^n ))/(√x))dx=?

$$\int\frac{\left(\mathrm{x}^{\mathrm{m}} −\mathrm{x}^{\mathrm{n}} \right)}{\sqrt{\mathrm{x}}}\mathrm{dx}=? \\ $$

Question Number 101277    Answers: 2   Comments: 0

∫_0 ^1 (((x−1) dx )/((x+1)ln (x)))

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

Question Number 101192    Answers: 1   Comments: 2

∫ (x/(1+sin x)) dx

$$\int\:\frac{{x}}{\mathrm{1}+\mathrm{sin}\:{x}}\:{dx}\: \\ $$

Question Number 101178    Answers: 2   Comments: 0

∫ ((((√x)−x)^2 )/x^2 ) dx ?

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

Question Number 101073    Answers: 1   Comments: 0

∫_0 ^∞ ((sin(logx))/(logx))dx

$$\int_{\mathrm{0}} ^{\infty} \frac{{sin}\left({logx}\right)}{{logx}}{dx} \\ $$

Question Number 101057    Answers: 1   Comments: 0

Find the area bounded the curves f(x)= ∣x^3 −4x^2 +3x∣ and x−axis

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{area}\:\mathrm{bounded}\:\mathrm{the}\: \\ $$$$\mathrm{curves}\:\mathrm{f}\left(\mathrm{x}\right)=\:\mid{x}^{\mathrm{3}} −\mathrm{4}{x}^{\mathrm{2}} +\mathrm{3}{x}\mid\:\mathrm{and}\: \\ $$$$\mathrm{x}−\mathrm{axis}\: \\ $$

Question Number 101079    Answers: 2   Comments: 2

Question Number 101023    Answers: 0   Comments: 0

∫tan^(1/5) x cotx secxdx

$$\int{tan}^{\frac{\mathrm{1}}{\mathrm{5}}} {x}\:{cotx}\:{secxdx} \\ $$

Question Number 101014    Answers: 0   Comments: 0

Show that ∫_(−∞) ^(+∞) (dx/(1+(x+tanx)^2 )) = π

$${Show}\:{that} \\ $$$$\int_{−\infty} ^{+\infty} \frac{{dx}}{\mathrm{1}+\left({x}+{tanx}\right)^{\mathrm{2}} }\:\:\:=\:\:\:\pi \\ $$

Question Number 101011    Answers: 0   Comments: 5

∫_0 ^∞ ((sinx)/x)dx

$$\int_{\mathrm{0}} ^{\infty} \frac{{sinx}}{{x}}{dx} \\ $$

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

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