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IntegrationQuestion and Answers: Page 131

Question Number 116846 Answers: 0 Comments: 2

...nice calculus... prove that :: ∫_0 ^( (π/2)) (√(((2^x −1)sin^3 (x))/((2^x +1)(sin^3 (x)+cos^3 (x))))) dx<(π/8) ...m.n.1970...

$$\:\:\:\:\:\:\:\:\:\:\:\:\:...{nice}\:\:{calculus}... \\ $$$$\:\:\:{prove}\:\:{that}\::: \\ $$$$ \\ $$$$\:\: \\ $$$$\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \sqrt{\frac{\left(\mathrm{2}^{{x}} −\mathrm{1}\right){sin}^{\mathrm{3}} \left({x}\right)}{\left(\mathrm{2}^{{x}} +\mathrm{1}\right)\left({sin}^{\mathrm{3}} \left({x}\right)+{cos}^{\mathrm{3}} \left({x}\right)\right)}}\:\:{dx}<\frac{\pi}{\mathrm{8}} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:...{m}.{n}.\mathrm{1970}... \\ $$$$ \\ $$

Question Number 116844 Answers: 1 Comments: 0

∫ ((8x+sin^(−1) (2x))/( (√(1−4x^2 )))) dx

$$\int\:\frac{\mathrm{8x}+\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{2x}\right)}{\:\sqrt{\mathrm{1}−\mathrm{4x}^{\mathrm{2}} }}\:\mathrm{dx}\: \\ $$

Question Number 116815 Answers: 2 Comments: 0

∫ (dx/((x−2)(x^2 +4))) =?

$$\int\:\frac{\mathrm{dx}}{\left(\mathrm{x}−\mathrm{2}\right)\left(\mathrm{x}^{\mathrm{2}} +\mathrm{4}\right)}\:=? \\ $$

Question Number 116813 Answers: 2 Comments: 0

∫_1 ^(√3) ((√(1+x^2 ))/x^2 ) dx ?

$$\:\:\:\:\:\:\underset{\mathrm{1}} {\overset{\sqrt{\mathrm{3}}} {\int}}\:\frac{\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }}{\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx}\:? \\ $$

Question Number 116803 Answers: 0 Comments: 0

Solve for X(x,y,z), Y(x,y,z), Z(x,y,z) { (((∂Z/∂y)−(∂Y/∂z)=1−x^2 )),(((∂Z/∂x)−(∂X/∂z)=−(y^2 /2))),(((∂Y/∂x)−(∂X/∂y)=z(2x−y))) :} where { ((X(x,y,0)=0)),((Y(x,y,0)=0)),((Z(x,y,z)=0)) :}

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{X}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right),\:\mathrm{Y}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right),\:\mathrm{Z}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right) \\ $$$$\begin{cases}{\frac{\partial\mathrm{Z}}{\partial\mathrm{y}}−\frac{\partial\mathrm{Y}}{\partial\mathrm{z}}=\mathrm{1}−\mathrm{x}^{\mathrm{2}} }\\{\frac{\partial\mathrm{Z}}{\partial\mathrm{x}}−\frac{\partial\mathrm{X}}{\partial\mathrm{z}}=−\frac{\mathrm{y}^{\mathrm{2}} }{\mathrm{2}}}\\{\frac{\partial\mathrm{Y}}{\partial\mathrm{x}}−\frac{\partial\mathrm{X}}{\partial\mathrm{y}}=\mathrm{z}\left(\mathrm{2x}−\mathrm{y}\right)}\end{cases}\:\mathrm{where}\:\begin{cases}{\mathrm{X}\left(\mathrm{x},\mathrm{y},\mathrm{0}\right)=\mathrm{0}}\\{\mathrm{Y}\left(\mathrm{x},\mathrm{y},\mathrm{0}\right)=\mathrm{0}}\\{\mathrm{Z}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)=\mathrm{0}}\end{cases} \\ $$

Question Number 116806 Answers: 2 Comments: 0

Hi Prove that: ∫_(-∞) ^(+∞) -e^(-x^2 ) dx=(√π) Thanks beforehand

$$\mathrm{Hi} \\ $$$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\:\int_{-\infty} ^{+\infty} -\mathrm{e}^{-\mathrm{x}^{\mathrm{2}} } \mathrm{dx}=\sqrt{\pi} \\ $$$$\mathrm{Thanks}\:\mathrm{beforehand} \\ $$$$ \\ $$

Question Number 116796 Answers: 1 Comments: 0

... calculus ... prove that :: ∫_0 ^( 1) (((1−x^p )(1−x^q )x^(r−1) )/(log(x)))dx=^(???) log( (((p+q+r+1)r)/((p+r)(q+r))) ) m.n.1970

$$\:\:\:\:\:\:\:\:...\:\:\:\:\:\:{calculus}\:\:... \\ $$$$ \\ $$$$\:\:\:\:\:{prove}\:\:{that}\::: \\ $$$$ \\ $$$$\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\left(\mathrm{1}−{x}^{{p}} \right)\left(\mathrm{1}−{x}^{{q}} \right){x}^{{r}−\mathrm{1}} }{{log}\left({x}\right)}{dx}\overset{???} {=}{log}\left(\:\frac{\left({p}+{q}+{r}+\mathrm{1}\right){r}}{\left({p}+{r}\right)\left({q}+{r}\right)}\:\right) \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:{m}.{n}.\mathrm{1970} \\ $$$$\: \\ $$

Question Number 116744 Answers: 4 Comments: 0

... (( nice)/(calculus)) ... prove that :: ∫_(−1) ^( ∞) (e^(−4x) /( (√(x+1)))) dx =((√π)/2) e^4 ... m.n .1970...

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\:\frac{\:{nice}}{{calculus}}\:... \\ $$$$\:\:{prove}\:\:{that}\::: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{−\mathrm{1}} ^{\:\infty} \frac{{e}^{−\mathrm{4}{x}} }{\:\sqrt{{x}+\mathrm{1}}}\:{dx}\:=\frac{\sqrt{\pi}}{\mathrm{2}}\:{e}^{\mathrm{4}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\:{m}.{n}\:.\mathrm{1970}... \\ $$$$ \\ $$

Question Number 116742 Answers: 0 Comments: 0

the curve y=f(x) is rotated about the x−axis to form solid.the volume of this solid is 0.5π(a−2sina cosa) for the limit of 0≤x≤a. find the value of a

$${the}\:{curve}\:{y}={f}\left({x}\right)\:{is}\:{rotated}\:{about}\:{the} \\ $$$${x}−{axis}\:{to}\:{form}\:{solid}.{the}\:{volume}\:{of}\:{this} \\ $$$${solid}\:{is}\:\mathrm{0}.\mathrm{5}\pi\left({a}−\mathrm{2}{sina}\:{cosa}\right)\:{for}\:{the}\:{limit} \\ $$$${of}\:\mathrm{0}\leqslant{x}\leqslant{a}.\:{find}\:{the}\:{value}\:{of}\:{a} \\ $$$$ \\ $$

Question Number 116738 Answers: 1 Comments: 1

determine the area of the region bounded by y=(2x+6)^(0.5 ) and y=x−1

$${determine}\:{the}\:{area}\:{of}\:{the}\:{region}\:{bounded} \\ $$$${by}\:{y}=\left(\mathrm{2}{x}+\mathrm{6}\right)^{\mathrm{0}.\mathrm{5}\:} {and}\:{y}={x}−\mathrm{1} \\ $$

Question Number 116737 Answers: 1 Comments: 0

find the lenght of the function y=sinx for 0<x<π

$${find}\:{the}\:{lenght}\:{of}\:{the}\:{function}\:{y}={sinx}\: \\ $$$${for}\:\mathrm{0}<{x}<\pi \\ $$$$ \\ $$

Question Number 116710 Answers: 2 Comments: 0

Question Number 116701 Answers: 2 Comments: 2

∫ (dx/(x+x(√x))) =?

$$\int\:\frac{\mathrm{dx}}{\mathrm{x}+\mathrm{x}\sqrt{\mathrm{x}}}\:=? \\ $$

Question Number 116687 Answers: 2 Comments: 0

Help please, to solve this ... If f(x)=1+x^2 for x∈[−2,2] and f(x)=5 otherwise. Then what is the value of ∫_(−2) ^(+2) f(2x^2 )dx?

$${Help}\:{please},\:{to}\:{solve}\:{this}\:... \\ $$$${If}\:{f}\left({x}\right)=\mathrm{1}+{x}^{\mathrm{2}} \:\:{for}\:{x}\in\left[−\mathrm{2},\mathrm{2}\right]\:{and}\: \\ $$$$\:\:\:\:\:\:{f}\left({x}\right)=\mathrm{5}\:\:\:\:\:\:\:\:{otherwise}. \\ $$$${Then}\:{what}\:{is}\:{the}\:{value}\:{of} \\ $$$$\int_{−\mathrm{2}} ^{+\mathrm{2}} {f}\left(\mathrm{2}{x}^{\mathrm{2}} \right){dx}? \\ $$$$ \\ $$

Question Number 116672 Answers: 2 Comments: 0

... nice calculus ... prove that : I = ∫_0 ^( 1) ((π/4) −Arctan(x))(dx/(1−x^2 )) = (G/2) ✓ G is catalan constant ... M.N.1970

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\:\:\:{nice}\:\:{calculus}\:... \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{prove}\:\:{that}\:: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{I}\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \left(\frac{\pi}{\mathrm{4}}\:\:−\mathscr{A}{rctan}\left({x}\right)\right)\frac{{dx}}{\mathrm{1}−{x}^{\mathrm{2}} }\:=\:\frac{\mathrm{G}}{\mathrm{2}}\:\:\checkmark\:\:\:\: \\ $$$$\:\:\:\:\:\:\: \\ $$$$\mathrm{G}\:{is}\:\:\:{catalan}\:\:{constant}\:... \\ $$$$\:\:\:\:\:\:\:\mathscr{M}.\mathscr{N}.\mathrm{1970} \\ $$$$ \\ $$$$ \\ $$$$\:\:\: \\ $$

Question Number 116667 Answers: 1 Comments: 0

... nice calculus... very nice integral:: demonstrate::: Ω = ∫_0 ^( 1) ((1−x)/((1+x+x^2 +x^3 )log(x))) dx=^(???) log((√(1/2))) .m.n.1970.

$$\:\:\:\:\:\:\:\:\:\:\:\:...\:\:\:{nice}\:\:{calculus}... \\ $$$$ \\ $$$$\:\:\:\:\:\:{very}\:{nice}\:\:{integral}:: \\ $$$$\:\:\:\:\:\:\:{demonstrate}::: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\Omega\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{1}−{x}}{\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} +{x}^{\mathrm{3}} \right){log}\left({x}\right)}\:{dx}\overset{???} {=}{log}\left(\sqrt{\frac{\mathrm{1}}{\mathrm{2}}}\right) \\ $$$$\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:.{m}.{n}.\mathrm{1970}. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\: \\ $$

Question Number 116763 Answers: 0 Comments: 1

∫(dx/( ((1+x^3 ))^(1/3) ))=?

$$ \\ $$$$\:\:\:\:\:\int\frac{{dx}}{\:\sqrt[{\mathrm{3}}]{\mathrm{1}+{x}^{\mathrm{3}} }}=? \\ $$

Question Number 116717 Answers: 2 Comments: 0

∫xdx

$$\int{xdx} \\ $$

Question Number 116627 Answers: 2 Comments: 0

... nice calculus... please evaluate :: Φ = (((∫_0 ^( (π/2)) ((e^x +e^(−x) )/(sin(x)+cos(x)))dx)^2 )/((∫_0 ^( (π/2)) (e^x /(sin(x)+cos(x)))dx)(∫_0 ^( (π/2)) (e^(−x) /(sin(x)+cos(x)))dx))) =??? m.n.1970

$$\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:...\:\:{nice}\:\:\:{calculus}... \\ $$$$ \\ $$$$\:\:\:\:\:\:{please}\:\:\:{evaluate}\::: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Phi\:=\:\frac{\left(\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{{e}^{{x}} +{e}^{−{x}} }{{sin}\left({x}\right)+{cos}\left({x}\right)}{dx}\right)^{\mathrm{2}} }{\left(\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{{e}^{{x}} }{{sin}\left({x}\right)+{cos}\left({x}\right)}{dx}\right)\left(\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{{e}^{−{x}} }{{sin}\left({x}\right)+{cos}\left({x}\right)}{dx}\right)}\:=???\:\:\:\: \\ $$$$ \\ $$$$\:\:\:\:{m}.{n}.\mathrm{1970} \\ $$$$ \\ $$

Question Number 116626 Answers: 2 Comments: 0

Π_(n=1) ^∞ (((a^2 n^2 )/((an)^2 −1))))=???

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\left(\frac{{a}^{\mathrm{2}} {n}^{\mathrm{2}} }{\left.\left({an}\right)^{\mathrm{2}} −\mathrm{1}\right)}\right)=??? \\ $$$$ \\ $$

Question Number 116590 Answers: 2 Comments: 0

∫ ((√x)/(1+x^3 )) dx =?

$$\:\int\:\frac{\sqrt{\mathrm{x}}}{\mathrm{1}+\mathrm{x}^{\mathrm{3}} }\:\mathrm{dx}\:=? \\ $$

Question Number 116586 Answers: 0 Comments: 0

1) explicite ∫_0 ^∞ ((arctan(1+x(2+t^2 )))/(2+t^2 ))dt withx>0 2)determine values of ∫_0 ^∞ ((arctan(3+t^2 ))/(2+t^2 ))dt and ∫_0 ^∞ ((arctan(5+2t^2 ))/(2+t^2 ))dt

$$\left.\mathrm{1}\right)\:{explicite}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{arctan}\left(\mathrm{1}+{x}\left(\mathrm{2}+{t}^{\mathrm{2}} \right)\right)}{\mathrm{2}+{t}^{\mathrm{2}} }{dt} \\ $$$${withx}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right){determine}\:{values}\:{of} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\frac{{arctan}\left(\mathrm{3}+{t}^{\mathrm{2}} \right)}{\mathrm{2}+{t}^{\mathrm{2}} }{dt}\:{and}\: \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\frac{{arctan}\left(\mathrm{5}+\mathrm{2}{t}^{\mathrm{2}} \right)}{\mathrm{2}+{t}^{\mathrm{2}} }{dt} \\ $$

Question Number 116564 Answers: 2 Comments: 0

∫ (dx/(x+(x)^(1/(3 )) )) ?

$$\int\:\frac{\mathrm{dx}}{\mathrm{x}+\sqrt[{\mathrm{3}\:}]{\mathrm{x}}}\:? \\ $$

Question Number 116560 Answers: 1 Comments: 1

if arctan(x+iy) =a+ib with a and b reals determine a and b

$${if}\:{arctan}\left({x}+{iy}\right)\:={a}+{ib} \\ $$$${with}\:{a}\:{and}\:{b}\:{reals}\:{determine} \\ $$$${a}\:{and}\:{b} \\ $$

Question Number 116557 Answers: 2 Comments: 0

calculate ∫_0 ^∞ ((ln(1+x(1+t^2 ))/(1+t^2 )) dt with x>0 2) find the value of ∫_0 ^∞ ((ln(2+t^2 ))/(1+t^2 ))dt and ∫_0 ^∞ ((ln(3+2t^2 ))/(1+t^2 ))dt

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}\left(\mathrm{1}+{x}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)\right.}{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt} \\ $$$${with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}\left(\mathrm{2}+{t}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$$${and}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}\left(\mathrm{3}+\mathrm{2}{t}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$

Question Number 116554 Answers: 1 Comments: 0

study the integral ∫_0 ^∞ ((sin^n x)/x^n )dx n integr and n≥2

$${study}\:{the}\:{integral}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{sin}^{{n}} {x}}{{x}^{{n}} }{dx} \\ $$$${n}\:{integr}\:{and}\:{n}\geqslant\mathrm{2} \\ $$

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