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Question Number 220963 Answers: 1 Comments: 0
$${Let}\:{f}:\mathbb{R}^{\mathrm{2}} \rightarrow\mathbb{R}\:{be}\:{defined}\:{by}\:{f}\left({x},{y}\right)=\left\{\frac{{y}}{\underset{\:\:\mathrm{1},\:{y}=\mathrm{0}} {\mathrm{sin}\:{y}}},\:{y}\neq\mathrm{0}\right. \\ $$$${Then}\:{the}\:{integral}\:\frac{\mathrm{1}}{\pi^{\mathrm{2}} }\underset{{x}=\mathrm{0}} {\overset{\mathrm{1}} {\int}}\underset{{y}=\mathrm{sin}^{−\mathrm{1}} {x}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}{f}\left({x},{y}\right){dy}\:{dx}\:{correct}\:{upto}\:{three}\:{decimal}\:{places},{is}... \\ $$
Question Number 220950 Answers: 1 Comments: 0
$$ \\ $$$$\:\:\:\:\int_{\:\mathrm{0}} ^{\:\pi} \int_{\:\mathrm{0}} ^{\:\mathrm{1}} \int_{\:\mathrm{0}} ^{\:\:\pi} \:\mathrm{sin}^{\:\mathrm{2}} \:{x}\:+\:{y}\:\mathrm{sin}\:{z}\:{dxdydz}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\:\pi\:\left(\mathrm{2}\:+\:\pi\right)\:\:\:\:\:\: \\ $$$$ \\ $$
Question Number 220948 Answers: 1 Comments: 0
$$\int\:{x}^{\mathrm{2}} \sqrt{\mathrm{5}−{x}^{\mathrm{6}} }{dx} \\ $$
Question Number 220904 Answers: 0 Comments: 0
$$ \\ $$$$\:\:\int\int\int_{\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{3}} } \frac{{x}^{\mathrm{4}} {y}^{\mathrm{3}} {z}^{\mathrm{2}} }{\left({x}+{y}+{z}\right)\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \right)−\left({x}^{\mathrm{3}} +{y}^{\mathrm{3}} +{z}^{\mathrm{3}} \right)}\:{dxdydz}\:\:\:\:\:\:\: \\ $$$$\: \\ $$
Question Number 220899 Answers: 0 Comments: 0
$$ \\ $$$$\:\:\:\int\int\int_{\left[\mathrm{0},\mathrm{1}\right]^{\:\mathrm{3}} } \:\frac{\mathrm{1}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{y}^{\mathrm{2}} \right)\left(\mathrm{1}+{z}^{\mathrm{2}} \right)\left(\mathrm{1}+{xyz}\right)}\:{dxdydz}\:\:\:\:\:\:\: \\ $$$$ \\ $$
Question Number 220898 Answers: 0 Comments: 0
$$ \\ $$$$\:\:\:\int\int\int_{\left[\mathrm{0},\mathrm{1}\right]^{\:\mathrm{3}} \:\:} \frac{\mathrm{1}}{\:\sqrt{\left(\mathrm{1}\:−{x}\right)\left(\mathrm{1}\:−\:{y}\right)\left(\mathrm{1}\:−{z}\right)\left(\mathrm{1}\:−\:{xyz}\right)}}\:{dxdydz}\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$
Question Number 220897 Answers: 0 Comments: 0
$$ \\ $$$$\:\:\:\int\int\int_{\:\left[\mathrm{0},\mathrm{1}\right]^{\:\mathrm{3}} } \:\frac{\mathrm{1}}{\mathrm{1}\:+\:{x}^{\mathrm{2}} {y}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} {z}^{\mathrm{2}} \:+\:{z}^{\mathrm{2}} {x}^{\mathrm{2}} }\:{dxdydz}\:\:\:\:\: \\ $$$$ \\ $$
Question Number 220896 Answers: 1 Comments: 0
$$ \\ $$$$\:\:\:\int\int\int_{\:\left[\mathrm{0},\infty\right]^{\:\mathrm{3}} } \frac{{x}^{\mathrm{2}} {y}^{\mathrm{2}} {z}^{\mathrm{2}} }{\left(\mathrm{1}\:+\:{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:+\:{z}^{\mathrm{2}} \right)^{\mathrm{5}} }\:{dxdydz}\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$
Question Number 220895 Answers: 0 Comments: 0
$$ \\ $$$$\:\:\:\int\int\int_{\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{3}} } \frac{{ln}\:\left(\mathrm{1}\:+\:{xyz}\right)}{\left(\mathrm{1}\:+\:{x}\right)\left(\mathrm{1}\:+\:{y}\right)\left(\mathrm{1}\:+\:{z}\right)}\:{dxdydz}\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$
Question Number 220892 Answers: 2 Comments: 0
$$ \\ $$$$\:\:\:\:\int\int\int_{\boldsymbol{{x}}^{\mathrm{2}} \:+\:\boldsymbol{{y}}^{\mathrm{2}} \:+\:\boldsymbol{{z}}^{\mathrm{2}} \:\:\leqslant\:\mathrm{1}} \:\frac{\mathrm{1}}{\left(\mathrm{1}\:+\:{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:+\:{z}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{dxdydz}\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$
Question Number 220891 Answers: 1 Comments: 0
$$ \\ $$$$\:\:\:\:\:\int\int\int_{\left[\mathrm{0},\infty\right]^{\:\mathrm{3}} } \:\frac{{e}^{−\left({x}\:+\:{y}\:+\:{z}\:\right)} }{\mathrm{1}\:+\:{xyz}}\:{dxdydz} \\ $$$$ \\ $$
Question Number 220889 Answers: 2 Comments: 0
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\int\int\int_{\:\left[\mathrm{0},\mathrm{1}\left[^{\:\mathrm{3}} \right.\right.} \:\frac{\mathrm{1}}{\mathrm{1}\:+\:{xyz}}\:{dxdydz} \\ $$$$ \\ $$
Question Number 220850 Answers: 1 Comments: 1
$$\int\sqrt{\frac{{x}+\mathrm{1}}{{x}+\mathrm{2}}}.\frac{\mathrm{1}}{{x}+\mathrm{3}}{dx} \\ $$
Question Number 220848 Answers: 1 Comments: 2
$$ \\ $$$$\:\:\:\:\:\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{1}}{\mathrm{2}}\:\sqrt{\frac{\left(\mathrm{2}\:−\:\mathrm{6}{x}\:+\:\mathrm{3}{x}^{\mathrm{2}} \right)^{\mathrm{2}} +\:\mathrm{4}\left(\mathrm{2}{x}\:−\:\mathrm{3}{x}^{\mathrm{2}} \:+\:{x}^{\mathrm{3}} \right)}{\mathrm{2}{x}\:−\:\mathrm{3}{x}^{\mathrm{2}} \:+\:{x}^{\mathrm{3}} }}\:{dx}\:\:\:\:\:\:\: \\ $$$$ \\ $$
Question Number 220842 Answers: 0 Comments: 2
$$ \\ $$$$\:\:\:\:\:\:\mathrm{J}=\int_{\mathrm{0}} ^{\:\infty} {e}^{−{t}} {u}_{\pi} \left({t}\right){cos}\left({t}\right){dt}=? \\ $$$$ \\ $$$$\:{note}:\:\:{u}_{{c}} \left({t}\right)=\:\begin{cases}{\:\mathrm{0}\:\:\:\:\:\:\:\:{t}<{c}}\\{\:\mathrm{1}\:\:\:\:\:\:\:\:\:{t}>{c}\:\:}\end{cases}\:\:;\:\:\:\:{c}\geqslant\mathrm{0} \\ $$
Question Number 220791 Answers: 0 Comments: 0
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\frac{{x}^{\mathrm{2}} }{\boldsymbol{\mathrm{sin}}\:{x}\:+\:\mathrm{1}}\:{dx} \\ $$$$ \\ $$
Question Number 220770 Answers: 1 Comments: 0
$$ \\ $$$$\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{\:{sin}\left({x}\right)}{\:\sqrt{\mathrm{1}\:+\sqrt{{sin}\left(\mathrm{2}{x}\right)}}}{dx}\: \\ $$
Question Number 220730 Answers: 2 Comments: 0
$$ \\ $$$$\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{1}}{\:\sqrt{{x}\left(\mathrm{1}\:−\:{x}\right)\left(\mathrm{1}\:+\:{kx}\right)}}\:{dx}\:,\:\left(−\mathrm{1}\:<\:{k}\:<\:\mathrm{1}\right)\:\:\: \\ $$$$ \\ $$
Question Number 220715 Answers: 2 Comments: 1
Question Number 220707 Answers: 2 Comments: 0
$$\underset{−\infty} {\overset{+\infty} {\int}}\frac{{x}\left(\mathrm{tan}^{−\mathrm{1}} \:{x}\right)^{\mathrm{3}} }{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} \left(\mathrm{1}+\mathrm{e}^{\mathrm{4tan}^{−\mathrm{1}} \:{x}} \right)}{dx}=? \\ $$
Question Number 220677 Answers: 1 Comments: 0
$$\int\sqrt{{x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{1}\:}}{dx} \\ $$
Question Number 220676 Answers: 1 Comments: 0
$$\int\:\frac{{xdx}}{\left(\mathrm{1}−{cosx}\right)^{\mathrm{2}} } \\ $$
Question Number 220674 Answers: 1 Comments: 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
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{1}} ^{\:\mathrm{2}} \:\frac{\mathrm{2}{x}^{\mathrm{2}} }{\:\sqrt{\left(\mathrm{2}{x}\:−\:\mathrm{1}\right)\centerdot\left(\mathrm{2}{x}\:+\:\mathrm{2}\right)}}\:{dx} \\ $$$$ \\ $$
Question Number 220544 Answers: 1 Comments: 0
$$ \\ $$$$\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{Prove}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{inequality}}; \\ $$$$\:\:\:\:\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{ln}\left(\mathrm{1}\:+\:{x}^{\mathrm{2}} \right)}{\mathrm{1}\:+\:{x}^{\mathrm{2}} }\:{dx}\:<\:\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\frac{{x}\:\mathrm{ln}\left(\mathrm{1}\:+\:{x}^{\mathrm{2}} \right)}{\mathrm{1}\:+\:{x}^{\mathrm{2}} \:}\:{dx}\:+\:\frac{\mathrm{1}}{\mathrm{3}}\:\:\:\:\:\:\:\: \\ $$$$ \\ $$
Question Number 220526 Answers: 2 Comments: 0
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