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IntegrationQuestion and Answers: Page 114
Question Number 128529 Answers: 1 Comments: 2
Question Number 128511 Answers: 1 Comments: 0
$$\:\mathcal{H}=\int\:\frac{\mathrm{2017x}^{\mathrm{2016}} +\mathrm{2018x}^{\mathrm{2017}} }{\mathrm{1}+\mathrm{x}^{\mathrm{4034}} +\mathrm{2x}^{\mathrm{4035}} +\mathrm{x}^{\mathrm{4036}} }\:\mathrm{dx}\: \\ $$
Question Number 128499 Answers: 1 Comments: 0
$$\mathrm{find}\:\:\mathrm{u}_{\mathrm{n}} =\int_{\mathrm{1}} ^{\infty} \:\:\frac{\left[\mathrm{ne}^{−\mathrm{x}} \right]}{\mathrm{n}^{\mathrm{3}} }\mathrm{dx} \\ $$
Question Number 128495 Answers: 0 Comments: 0
$$\mathrm{find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{sinx}}{\left[\mathrm{x}\right]}\mathrm{dx} \\ $$
Question Number 128471 Answers: 2 Comments: 0
$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\:\:{calculus}\:... \\ $$$$\:\:\:\:\:\:\:\Phi\overset{?} {=}\int_{\mathrm{0}} ^{\:\mathrm{1}} \left({ln}\left({x}\right)\right)^{\mathrm{2}} {ln}\left(\sqrt{−{ln}\left({x}\right)}\:{dx}\right. \\ $$
Question Number 128417 Answers: 1 Comments: 0
$$\:\:\:\:\:\:\:\:\:\eta\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\mathrm{x}^{\mathrm{3}} \:\left(\mathrm{1}−\mathrm{x}^{\mathrm{3}} \right)^{\mathrm{n}−\mathrm{1}} \:\mathrm{dx}\: \\ $$
Question Number 128408 Answers: 1 Comments: 0
$$\rho\:=\:\int\:\frac{\mathrm{sin}\:\left(\mathrm{4}{x}\right)}{\mathrm{sin}\:^{\mathrm{4}} \left({x}\right)+\mathrm{cos}\:^{\mathrm{4}} \left({x}\right)}\:{dx}\: \\ $$
Question Number 128385 Answers: 1 Comments: 0
$$\int_{\mathrm{1}} ^{\:\pi} \begin{vmatrix}{\mathrm{x}^{\mathrm{3}} }&{\mathrm{lnx}}&{\mathrm{sinx}}\\{\mathrm{3x}^{\mathrm{2}} }&{\frac{\mathrm{1}}{\mathrm{x}}}&{\mathrm{cosx}}\\{\mathrm{6}}&{\mathrm{2x}^{−\mathrm{3}} }&{−\mathrm{cosx}}\end{vmatrix}\mathrm{dx}\:=?\: \\ $$
Question Number 128373 Answers: 1 Comments: 0
$$\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{\frac{\mathrm{3}}{\mathrm{2}}} \left(\mathrm{1}−{x}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \:{dx} \\ $$
Question Number 128334 Answers: 1 Comments: 0
$$\Omega\:=\:\int\:\frac{\mathrm{x}^{\mathrm{2}} }{\:\sqrt{\left(\mathrm{a}+\mathrm{bx}^{\mathrm{2}} \right)^{\mathrm{5}} }}\:\mathrm{dx}\:;\:\mathrm{where}\::\:\mathrm{a};\:\mathrm{b}\:>\mathrm{0}\: \\ $$
Question Number 128346 Answers: 2 Comments: 0
Question Number 128316 Answers: 1 Comments: 0
$$\:\:\:\:\:\:\:\:\:\:\:\:{nice}\:\:{calculus} \\ $$$$\:\:\:\Omega=\:\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}^{\mathrm{3}} \left({x}\right)}{{x}^{\mathrm{2}} }{dx}=? \\ $$$$ \\ $$
Question Number 128285 Answers: 1 Comments: 0
$$\:\mathrm{cos}\:\left(\frac{\pi}{\mathrm{7}}\right)−\mathrm{cos}\:\left(\frac{\mathrm{2}\pi}{\mathrm{7}}\right)+\mathrm{cos}\:\left(\frac{\mathrm{3}\pi}{\mathrm{7}}\right)\:=? \\ $$
Question Number 128276 Answers: 1 Comments: 0
$$\:\int_{{e}^{\mathrm{2}} } ^{\:\infty} \:\frac{{dx}}{{x}^{\mathrm{3}} \:\mathrm{ln}\:{x}}\:? \\ $$
Question Number 128262 Answers: 2 Comments: 0
Question Number 128251 Answers: 1 Comments: 0
Question Number 128244 Answers: 0 Comments: 0
$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{nice}\:\:{calculus}... \\ $$$$\:\:\:\:\:{prove}\:\:{that}\:\:::\Omega=\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} {ln}\left({sin}\left({x}\right)\right){d}=\frac{−\pi}{\mathrm{4}}{log}\left(\mathrm{2}\right)−\frac{{G}}{\mathrm{2}} \\ $$$$\:\:\:\:{log}\left(\mathrm{2}{sin}\left({x}\right)\right)=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{−\mathrm{1}}{{n}}{cos}\left(\mathrm{2}{nx}\right) \\ $$$$\:\Omega=\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} \left\{−{log}\left(\mathrm{2}\right)−\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{cos}\left(\mathrm{2}{nx}\right)}{{n}}\right\}{dx} \\ $$$$=\frac{−\pi}{\mathrm{4}}{log}\left(\mathrm{2}\right)−\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} \frac{{cos}\left(\mathrm{2}{nx}\right)}{{n}}{dx} \\ $$$$\:=\frac{−\pi}{\mathrm{4}}{log}\left(\mathrm{2}\right)−\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left[\frac{\mathrm{1}}{\mathrm{2}{n}^{\mathrm{2}} }{sin}\left(\mathrm{2}{nx}\right)\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \\ $$$$\:\:=\frac{−\pi}{\mathrm{4}}{log}\left(\mathrm{2}\right)−\frac{\mathrm{1}}{\mathrm{2}}\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{sin}\left(\frac{{n}\pi}{\mathrm{2}}\right)}{{n}^{\mathrm{2}} } \\ $$$$\:=\frac{−\pi}{\mathrm{4}}{log}\left(\mathrm{2}\right)−\frac{\mathrm{1}}{\mathrm{2}}\left\{\frac{\mathrm{1}}{\mathrm{1}^{\mathrm{2}} }−\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{2}} }\:+\frac{\mathrm{1}}{\mathrm{5}^{\mathrm{2}} }−..\right\} \\ $$$$\:=\frac{−\pi}{\mathrm{4}}{log}\left(\mathrm{2}\right)−\frac{\mathrm{1}}{\mathrm{2}}\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} }{\left(\mathrm{2}{n}−\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\therefore\:\:\Omega\:=\frac{−\pi}{\mathrm{4}}{log}\left(\mathrm{2}\right)−\frac{{G}}{\mathrm{2}}\:\:\checkmark \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{G}:=\:{catalan}\:\:{constant}... \\ $$
Question Number 128221 Answers: 1 Comments: 0
$$\:{Given}\:{f}\left({x}+\frac{\mathrm{1}}{{x}}\right)\:=\:{x}^{\mathrm{4}} −\frac{\mathrm{1}}{{x}^{\mathrm{4}} }+\mathrm{2} \\ $$$$\:{then}\:\int_{\mathrm{1}} ^{\:\mathrm{2}} \left(\mathrm{1}−{x}^{−\mathrm{2}} \right){f}\left({x}\right){dx}= \\ $$
Question Number 128194 Answers: 1 Comments: 2
Question Number 128192 Answers: 0 Comments: 0
Question Number 128175 Answers: 1 Comments: 1
$$\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{{dx}\:{dy}}{\mathrm{1}−{xy}^{\mathrm{3}} }\:? \\ $$
Question Number 128109 Answers: 1 Comments: 0
$$\emptyset\:=\:\int\:\left(\mathrm{x}−\mathrm{2}\right)\:\sqrt{\frac{\mathrm{x}+\mathrm{1}}{\mathrm{x}−\mathrm{1}}}\:\mathrm{dx}\: \\ $$
Question Number 128082 Answers: 0 Comments: 0
$$\beta\:=\:\int\:\frac{\mathrm{1}+\mathrm{ln}\:\left(\mathrm{x}\right)}{\mathrm{x}.\mathrm{cos}\:^{\mathrm{2}} \left(\mathrm{x}\right)}\:\mathrm{dx}\: \\ $$
Question Number 128079 Answers: 1 Comments: 0
$$\Omega\:=\:\int\:\mathrm{ln}\:\left(\mathrm{x}+\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{a}^{\mathrm{2}} }\:\right)\mathrm{dx} \\ $$
Question Number 128073 Answers: 1 Comments: 0
$$\int\:\frac{\mathrm{sin}\left(\mathrm{2x}\right)}{\left(\mathrm{1}\:\:−\:\:\mathrm{x}\right)^{\mathrm{3}} }\:\:\mathrm{dx} \\ $$
Question Number 128057 Answers: 3 Comments: 0
$$\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{1}}{\mathrm{1}−\mathrm{xy}^{\mathrm{2}} }\:\mathrm{dx}\:\mathrm{dy}\:=? \\ $$
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