Question and Answers Forum

IntegrationQuestion and Answers: Page 1

Question Number 220963 Answers: 1 Comments: 0

Let f:R^2 →R be defined by f(x,y)={(y/(sin y_( 1, y=0) )), y≠0 Then the integral (1/π^2 )∫_(x=0) ^1 ∫_(y=sin^(−1) x) ^(π/2) f(x,y)dy dx correct upto three decimal places,is...

$${Let}\:{f}:\mathbb{R}^{\mathrm{2}} \rightarrow\mathbb{R}\:{be}\:{defined}\:{by}\:{f}\left({x},{y}\right)=\left\{\frac{{y}}{\underset{\:\:\mathrm{1},\:{y}=\mathrm{0}} {\mathrm{sin}\:{y}}},\:{y}\neq\mathrm{0}\right. \\ $$$${Then}\:{the}\:{integral}\:\frac{\mathrm{1}}{\pi^{\mathrm{2}} }\underset{{x}=\mathrm{0}} {\overset{\mathrm{1}} {\int}}\underset{{y}=\mathrm{sin}^{−\mathrm{1}} {x}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}{f}\left({x},{y}\right){dy}\:{dx}\:{correct}\:{upto}\:{three}\:{decimal}\:{places},{is}... \\ $$

Question Number 220950 Answers: 1 Comments: 0

∫_( 0) ^( π) ∫_( 0) ^( 1) ∫_( 0) ^( π) sin^( 2) x + y sin z dxdydz = (1/2) π (2 + π)

$$ \\ $$$$\:\:\:\:\int_{\:\mathrm{0}} ^{\:\pi} \int_{\:\mathrm{0}} ^{\:\mathrm{1}} \int_{\:\mathrm{0}} ^{\:\:\pi} \:\mathrm{sin}^{\:\mathrm{2}} \:{x}\:+\:{y}\:\mathrm{sin}\:{z}\:{dxdydz}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\:\pi\:\left(\mathrm{2}\:+\:\pi\right)\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 220948 Answers: 1 Comments: 0

∫ x^2 (√(5−x^6 ))dx

$$\int\:{x}^{\mathrm{2}} \sqrt{\mathrm{5}−{x}^{\mathrm{6}} }{dx} \\ $$

Question Number 220904 Answers: 0 Comments: 0

∫∫∫_([0,1]^3 ) ((x^4 y^3 z^2 )/((x+y+z)(x^2 +y^2 +z^2 )−(x^3 +y^3 +z^3 ))) dxdydz

$$ \\ $$$$\:\:\int\int\int_{\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{3}} } \frac{{x}^{\mathrm{4}} {y}^{\mathrm{3}} {z}^{\mathrm{2}} }{\left({x}+{y}+{z}\right)\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \right)−\left({x}^{\mathrm{3}} +{y}^{\mathrm{3}} +{z}^{\mathrm{3}} \right)}\:{dxdydz}\:\:\:\:\:\:\: \\ $$$$\: \\ $$

Question Number 220899 Answers: 0 Comments: 0

∫∫∫_([0,1]^( 3) ) (1/((1+x^2 )(1+y^2 )(1+z^2 )(1+xyz))) dxdydz

$$ \\ $$$$\:\:\:\int\int\int_{\left[\mathrm{0},\mathrm{1}\right]^{\:\mathrm{3}} } \:\frac{\mathrm{1}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{y}^{\mathrm{2}} \right)\left(\mathrm{1}+{z}^{\mathrm{2}} \right)\left(\mathrm{1}+{xyz}\right)}\:{dxdydz}\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 220898 Answers: 0 Comments: 0

∫∫∫_([0,1]^( 3) ) (1/( (√((1 −x)(1 − y)(1 −z)(1 − xyz))))) dxdydz

$$ \\ $$$$\:\:\:\int\int\int_{\left[\mathrm{0},\mathrm{1}\right]^{\:\mathrm{3}} \:\:} \frac{\mathrm{1}}{\:\sqrt{\left(\mathrm{1}\:−{x}\right)\left(\mathrm{1}\:−\:{y}\right)\left(\mathrm{1}\:−{z}\right)\left(\mathrm{1}\:−\:{xyz}\right)}}\:{dxdydz}\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 220897 Answers: 0 Comments: 0

∫∫∫_( [0,1]^( 3) ) (1/(1 + x^2 y^2 + y^2 z^2 + z^2 x^2 )) dxdydz

$$ \\ $$$$\:\:\:\int\int\int_{\:\left[\mathrm{0},\mathrm{1}\right]^{\:\mathrm{3}} } \:\frac{\mathrm{1}}{\mathrm{1}\:+\:{x}^{\mathrm{2}} {y}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} {z}^{\mathrm{2}} \:+\:{z}^{\mathrm{2}} {x}^{\mathrm{2}} }\:{dxdydz}\:\:\:\:\: \\ $$$$ \\ $$

Question Number 220896 Answers: 1 Comments: 0

∫∫∫_( [0,∞]^( 3) ) ((x^2 y^2 z^2 )/((1 + x^2 + y^2 + z^2 )^5 )) dxdydz

$$ \\ $$$$\:\:\:\int\int\int_{\:\left[\mathrm{0},\infty\right]^{\:\mathrm{3}} } \frac{{x}^{\mathrm{2}} {y}^{\mathrm{2}} {z}^{\mathrm{2}} }{\left(\mathrm{1}\:+\:{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:+\:{z}^{\mathrm{2}} \right)^{\mathrm{5}} }\:{dxdydz}\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 220895 Answers: 0 Comments: 0

∫∫∫_([0,1]^3 ) ((ln (1 + xyz))/((1 + x)(1 + y)(1 + z))) dxdydz

$$ \\ $$$$\:\:\:\int\int\int_{\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{3}} } \frac{{ln}\:\left(\mathrm{1}\:+\:{xyz}\right)}{\left(\mathrm{1}\:+\:{x}\right)\left(\mathrm{1}\:+\:{y}\right)\left(\mathrm{1}\:+\:{z}\right)}\:{dxdydz}\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 220892 Answers: 2 Comments: 0

∫∫∫_(x^2 + y^2 + z^2 ≤ 1) (1/((1 + x^2 +y^2 + z^2 )^2 )) dxdydz

$$ \\ $$$$\:\:\:\:\int\int\int_{\boldsymbol{{x}}^{\mathrm{2}} \:+\:\boldsymbol{{y}}^{\mathrm{2}} \:+\:\boldsymbol{{z}}^{\mathrm{2}} \:\:\leqslant\:\mathrm{1}} \:\frac{\mathrm{1}}{\left(\mathrm{1}\:+\:{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:+\:{z}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{dxdydz}\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 220891 Answers: 1 Comments: 0

∫∫∫_([0,∞]^( 3) ) (e^(−(x + y + z )) /(1 + xyz)) dxdydz

$$ \\ $$$$\:\:\:\:\:\int\int\int_{\left[\mathrm{0},\infty\right]^{\:\mathrm{3}} } \:\frac{{e}^{−\left({x}\:+\:{y}\:+\:{z}\:\right)} }{\mathrm{1}\:+\:{xyz}}\:{dxdydz} \\ $$$$ \\ $$

Question Number 220889 Answers: 2 Comments: 0

∫∫∫_( [0,1[^( 3) ) (1/(1 + xyz)) dxdydz

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\int\int\int_{\:\left[\mathrm{0},\mathrm{1}\left[^{\:\mathrm{3}} \right.\right.} \:\frac{\mathrm{1}}{\mathrm{1}\:+\:{xyz}}\:{dxdydz} \\ $$$$ \\ $$

Question Number 220850 Answers: 1 Comments: 1

∫(√((x+1)/(x+2))).(1/(x+3))dx

$$\int\sqrt{\frac{{x}+\mathrm{1}}{{x}+\mathrm{2}}}.\frac{\mathrm{1}}{{x}+\mathrm{3}}{dx} \\ $$

Question Number 220848 Answers: 1 Comments: 2

∫_( 0) ^( 1) (1/2) (√(((2 − 6x + 3x^2 )^2 + 4(2x − 3x^2 + x^3 ))/(2x − 3x^2 + x^3 ))) dx

$$ \\ $$$$\:\:\:\:\:\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{1}}{\mathrm{2}}\:\sqrt{\frac{\left(\mathrm{2}\:−\:\mathrm{6}{x}\:+\:\mathrm{3}{x}^{\mathrm{2}} \right)^{\mathrm{2}} +\:\mathrm{4}\left(\mathrm{2}{x}\:−\:\mathrm{3}{x}^{\mathrm{2}} \:+\:{x}^{\mathrm{3}} \right)}{\mathrm{2}{x}\:−\:\mathrm{3}{x}^{\mathrm{2}} \:+\:{x}^{\mathrm{3}} }}\:{dx}\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 220842 Answers: 0 Comments: 2

J=∫_0 ^( ∞) e^(−t) u_π (t)cos(t)dt=? note: u_c (t)= { (( 0 t<c)),(( 1 t>c )) :} ; c≥0

$$ \\ $$$$\:\:\:\:\:\:\mathrm{J}=\int_{\mathrm{0}} ^{\:\infty} {e}^{−{t}} {u}_{\pi} \left({t}\right){cos}\left({t}\right){dt}=? \\ $$$$ \\ $$$$\:{note}:\:\:{u}_{{c}} \left({t}\right)=\:\begin{cases}{\:\mathrm{0}\:\:\:\:\:\:\:\:{t}<{c}}\\{\:\mathrm{1}\:\:\:\:\:\:\:\:\:{t}>{c}\:\:}\end{cases}\:\:;\:\:\:\:{c}\geqslant\mathrm{0} \\ $$

Question Number 220791 Answers: 0 Comments: 0

∫_( 0) ^( 1) (x^2 /(sin x + 1)) dx

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\frac{{x}^{\mathrm{2}} }{\boldsymbol{\mathrm{sin}}\:{x}\:+\:\mathrm{1}}\:{dx} \\ $$$$ \\ $$

Question Number 220770 Answers: 1 Comments: 0

∫_0 ^( (π/2)) (( sin(x))/( (√(1 +(√(sin(2x)))))))dx

$$ \\ $$$$\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{\:{sin}\left({x}\right)}{\:\sqrt{\mathrm{1}\:+\sqrt{{sin}\left(\mathrm{2}{x}\right)}}}{dx}\: \\ $$

Question Number 220730 Answers: 2 Comments: 0

∫_0 ^( 1) (1/( (√(x(1 − x)(1 + kx))))) dx , (−1 < k < 1)

$$ \\ $$$$\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{1}}{\:\sqrt{{x}\left(\mathrm{1}\:−\:{x}\right)\left(\mathrm{1}\:+\:{kx}\right)}}\:{dx}\:,\:\left(−\mathrm{1}\:<\:{k}\:<\:\mathrm{1}\right)\:\:\: \\ $$$$ \\ $$

Question Number 220715 Answers: 2 Comments: 1

Question Number 220707 Answers: 2 Comments: 0

∫_(−∞) ^(+∞) ((x(tan^(−1) x)^3 )/((x^2 +1)^2 (1+e^(4tan^(−1) x) )))dx=?

$$\underset{−\infty} {\overset{+\infty} {\int}}\frac{{x}\left(\mathrm{tan}^{−\mathrm{1}} \:{x}\right)^{\mathrm{3}} }{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} \left(\mathrm{1}+\mathrm{e}^{\mathrm{4tan}^{−\mathrm{1}} \:{x}} \right)}{dx}=? \\ $$

Question Number 220677 Answers: 1 Comments: 0

∫(√(x+(√(x^2 +1 ))))dx

$$\int\sqrt{{x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{1}\:}}{dx} \\ $$

Question Number 220676 Answers: 1 Comments: 0

∫ ((xdx)/((1−cosx)^2 ))

$$\int\:\frac{{xdx}}{\left(\mathrm{1}−{cosx}\right)^{\mathrm{2}} } \\ $$

Question Number 220674 Answers: 1 Comments: 0

∫∫∫_( E ) (z^2 /( (√(x^2 + y^2 )))) dV with the boundaries of the integration region E defined by; • x^2 + y^2 + z^2 ≤ 4 • x^2 + y^2 ≥ 1 • z ≥ 0

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\int\int\int_{\:{E}\:} \:\:\frac{{z}^{\mathrm{2}} }{\:\sqrt{{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} }}\:\:{dV} \\ $$$$\:\:\:\:\:\mathrm{with}\:\mathrm{the}\:\mathrm{boundaries}\:\mathrm{of}\:\mathrm{the}\:\mathrm{integration}\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\mathrm{region}\:{E}\:\mathrm{defined}\:\mathrm{by};\: \\ $$$$\:\:\:\:\:\:\bullet\:{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} +\:{z}^{\mathrm{2}} \:\leqslant\:\mathrm{4} \\ $$$$\:\:\:\:\:\:\bullet\:{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:\geqslant\:\mathrm{1} \\ $$$$\:\:\:\:\:\:\bullet\:{z}\:\geqslant\:\mathrm{0} \\ $$$$ \\ $$

Question Number 220644 Answers: 2 Comments: 0

∫_1 ^( 2) ((2x^2 )/( (√((2x − 1)∙(2x + 2))))) dx

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{1}} ^{\:\mathrm{2}} \:\frac{\mathrm{2}{x}^{\mathrm{2}} }{\:\sqrt{\left(\mathrm{2}{x}\:−\:\mathrm{1}\right)\centerdot\left(\mathrm{2}{x}\:+\:\mathrm{2}\right)}}\:{dx} \\ $$$$ \\ $$

Question Number 220544 Answers: 1 Comments: 0

Prove that inequality; ∫_( 0) ^( 1) ((ln(1 + x^2 ))/(1 + x^2 )) dx < ∫_( 0) ^( 1) ((x ln(1 + x^2 ))/(1 + x^2 )) dx + (1/3)

$$ \\ $$$$\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{Prove}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{inequality}}; \\ $$$$\:\:\:\:\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{ln}\left(\mathrm{1}\:+\:{x}^{\mathrm{2}} \right)}{\mathrm{1}\:+\:{x}^{\mathrm{2}} }\:{dx}\:<\:\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\frac{{x}\:\mathrm{ln}\left(\mathrm{1}\:+\:{x}^{\mathrm{2}} \right)}{\mathrm{1}\:+\:{x}^{\mathrm{2}} \:}\:{dx}\:+\:\frac{\mathrm{1}}{\mathrm{3}}\:\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 220526 Answers: 2 Comments: 0

Pg 1 Pg 2 Pg 3 Pg 4 Pg 5 Pg 6 Pg 7 Pg 8 Pg 9 Pg 10

Contact: info@tinkutara.com