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Question Number 189266 Answers: 1 Comments: 0
$$\:\:\underset{\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\:\frac{\sqrt[{\mathrm{3}}]{\mathrm{tan}\:\mathrm{x}}}{\mathrm{1}+\mathrm{sin}\:\mathrm{2x}}\:\mathrm{dx}\:=? \\ $$
Question Number 189250 Answers: 0 Comments: 1
Question Number 189189 Answers: 1 Comments: 0
Question Number 189144 Answers: 0 Comments: 1
$$\: \\ $$$$\: \\ $$$$\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \int_{\mathrm{0}} ^{\:\mathrm{1}} \int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\sqrt{{x}\:+\:{y}\:+\:{z}}}{\:\sqrt{{x}}\:+\:\sqrt{{y}}\:+\:\sqrt{{z}}\:}\:{dxdydz} \\ $$$$\: \\ $$$$\: \\ $$
Question Number 189066 Answers: 2 Comments: 0
$$\:\mathrm{Given}\:\mathrm{f}\left(\mathrm{x}\right)+\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\left(\mathrm{x}+\mathrm{y}\right)^{\mathrm{2}} \:\mathrm{f}\left(\mathrm{y}\right)\:\mathrm{dy}=\mathrm{2x}^{\mathrm{2}} −\mathrm{3x}+\mathrm{1} \\ $$$$\:\mathrm{find}\:\mathrm{f}\left(\mathrm{x}\right). \\ $$
Question Number 189057 Answers: 0 Comments: 6
$$\: \\ $$$$\:\mathrm{Help}! \\ $$$$\: \\ $$$$\:\mathrm{Evaluate}\:\:\mathrm{the}\:\:\mathrm{following}\:\:\mathrm{integral}\:\:\mathrm{usings}\:\:\mathrm{Green}\:\mathrm{theorem}: \\ $$$$\: \\ $$$$\:\oint\mathrm{4xy}{d}\mathrm{x}\:\:+\:\:\mathrm{x}^{\mathrm{2}} {d}\mathrm{y} \\ $$$$\: \\ $$$$\:\mathrm{Where}\:\:{C}\:\:\mathrm{is}\:\:\mathrm{the}\:\:\mathrm{square}\:\:\mathrm{of}\:\:\mathrm{vertices}\:\:\left(\mathrm{0},\mathrm{0}\right),\:\left(\mathrm{0},\mathrm{2}\right),\:\left(\mathrm{2},\mathrm{0}\right)\:\:\mathrm{and}\:\:\left(\mathrm{2},\mathrm{2}\right). \\ $$$$\: \\ $$
Question Number 188982 Answers: 0 Comments: 2
Question Number 188889 Answers: 1 Comments: 0
$$ \\ $$$$\:\:{If},\:{y}=\:\frac{\:{Arcsin}\left(\sqrt{{x}}\:\right)}{\:\sqrt{\:{x}\:\left(\mathrm{1}−{x}\:\right)}}\:\:\Rightarrow \\ $$$$\:\:\:{y}'\:.{p}\left({x}\right)\:+\:{y}\:.{q}\left({x}\right)=\:\mathrm{1} \\ $$$$ \\ $$$$\:\:\:{find}\:,\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {p}\left({x}\right).{q}\left({x}\right){dx}=? \\ $$$$\:\:\:\:{p}\:,\:{q}\:\:{are}\:{two}\:{pllynomils}... \\ $$$$ \\ $$
Question Number 188861 Answers: 2 Comments: 0
Question Number 188552 Answers: 0 Comments: 1
Question Number 188511 Answers: 0 Comments: 0
$$\:\:\:{evaluate} \\ $$$$\int_{\mathrm{0}} ^{\pi} \frac{{dx}}{{a}+{b}\mathrm{cos}{x}\:}\:\:\:\:\:\:,\:\:\:{a}\:>\:\mathrm{0} \\ $$$$\:\:\:{and}\:{deduce}\:{that} \\ $$$$\:\:\:\int_{\mathrm{0}} ^{\pi} \frac{{dx}}{\left({a}+{b}\mathrm{cos}\:{x}\right)^{\mathrm{2}} }\:\:=\:\:\:\frac{\pi{a}}{\left({a}^{\mathrm{2}} −{b}^{\mathrm{2}} \right)^{\mathrm{3}/\mathrm{2}} }\:\:;\:\:{a}^{\mathrm{2}} >{b}^{\mathrm{2}} \\ $$$${and}\:\:\int_{\mathrm{0}} ^{\pi} \frac{\mathrm{cos}\:{x}\:{dx}}{\left({a}+{b}\mathrm{cos}\:{x}\right)^{\mathrm{2}} }\:\:=\:\frac{−\pi{b}}{\left({a}^{\mathrm{2}} −{b}^{\mathrm{2}} \right)^{\mathrm{3}/\mathrm{2}} }\:\:;\:\:{a}^{\mathrm{2}} >{b}^{\mathrm{2}} \\ $$
Question Number 188470 Answers: 0 Comments: 0
$$\underset{\mathrm{0}} {\overset{\pi/\mathrm{4}} {\int}}{arctan}\left(\sqrt{\frac{\mathrm{1}−{tan}^{\mathrm{2}} {x}}{\mathrm{2}}}\right){dx}\:=\:? \\ $$
Question Number 188384 Answers: 1 Comments: 0
$$\:\:\: \\ $$$$\:\:\:\mathrm{if}\:\:\:\:\int_{\mathrm{0}} ^{\infty} {e}^{−{ax}} {dx}\:\:=\:\:\frac{\mathrm{1}}{{a}} \\ $$$$\:\:\:\:\:\:\:\:{show}\:{that}\:\:\int_{\mathrm{0}} ^{\infty} {e}^{−{ax}} \:{x}^{{n}} {dx}\:\:=\:\:\frac{{n}!}{{a}^{{n}+\mathrm{1}} } \\ $$
Question Number 188381 Answers: 2 Comments: 0
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{Advanced}\:\:\mathrm{calculus} \\ $$$$ \\ $$$$\:\:\:\:\:\mathrm{Find}\:\:\mathrm{the}\:\:\mathrm{value}\:\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{series}. \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\Omega\:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\:\left(−\mathrm{1}\right)^{\:{n}} \:\zeta\:\left(\:{n}\:\right)}{{n}.\:\mathrm{2}^{\:{n}} }\:=\:? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\zeta\:\left(\:{z}\:\right)\:=\:\underset{\:{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:\mathrm{1}}{{n}^{\:{z}} \:}\:\:\:\:\:;\:\:\:\:\mathscr{R}{e}\:\left(\:{z}\:\right)>\mathrm{1} \\ $$$$ \\ $$
Question Number 188380 Answers: 1 Comments: 0
Question Number 188192 Answers: 2 Comments: 0
$$\:\:\: \\ $$$$\:\:\:\:{prove}\:{that} \\ $$$$\:\:\:\:\:\int_{\mathrm{0}} ^{\infty} {e}^{−{a}^{\mathrm{2}} {x}^{\mathrm{2}} } \mathrm{cos}\left(\mathrm{2}{bx}\right)\:{dx}\:\:\:=\:\:\:\frac{\sqrt{\pi}}{\mathrm{2}{a}}{e}^{−{b}^{\mathrm{2}} /{a}^{\mathrm{2}} } \\ $$$$ \\ $$$$ \\ $$
Question Number 188164 Answers: 1 Comments: 1
Question Number 188086 Answers: 1 Comments: 0
$$\:\:\:\: \\ $$$$\:\:\:\:\:\:\mathrm{I}\:\:\:=\:\:\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{tan}^{−\mathrm{1}} \left({x}/{a}\right)}{{x}\left({x}^{\mathrm{2}} +{b}^{\mathrm{2}} \right)}{dx} \\ $$
Question Number 188036 Answers: 1 Comments: 0
$$\int\mathrm{2}^{{x}} {e}^{{x}} {dx} \\ $$
Question Number 188035 Answers: 1 Comments: 0
$${solve} \\ $$$$\int\frac{{x}^{\mathrm{2}} }{\left({a}+{bx}\right)^{\mathrm{2}} }{dx} \\ $$
Question Number 188034 Answers: 1 Comments: 0
$${solve} \\ $$$$\int\frac{{x}^{\mathrm{2}} +\mathrm{3}}{{x}^{\mathrm{6}} \left({x}^{\mathrm{2}} +\mathrm{1}\right)}{dx} \\ $$
Question Number 187993 Answers: 0 Comments: 0
$${prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \sqrt[{\mathrm{3}}]{\frac{{sin}\mathrm{3}{x}}{{sin}\mathrm{2}{y}}}{dxdy}=\frac{\pi}{\mathrm{2}\sqrt{\mathrm{3}}} \\ $$
Question Number 187898 Answers: 2 Comments: 0
$$\:\:\int\:\frac{\mathrm{1}−\mathrm{cos}\:\mathrm{x}}{\mathrm{cos}\:\mathrm{x}+\mathrm{sin}\:\mathrm{x}−\mathrm{1}}\:\mathrm{dx}=? \\ $$
Question Number 187890 Answers: 1 Comments: 0
Question Number 187855 Answers: 2 Comments: 0
$${solve} \\ $$$$\int\frac{{x}^{\mathrm{4}} +{x}^{\mathrm{2}} +\mathrm{1}}{\mathrm{2}\left({x}^{\mathrm{2}} +\mathrm{1}\right)}{dx} \\ $$
Question Number 187703 Answers: 2 Comments: 0
$$\int\:\frac{\mathrm{1}}{\mathrm{5x}^{\mathrm{2}} \:\:−\:\:\mathrm{2x}\:\:−\:\:\mathrm{4}}\:\mathrm{dx} \\ $$
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