Question Number 121857 by Bird last updated on 12/Nov/20 $${find}\:\int_{\mathrm{0}} ^{\infty} \:{xe}^{−{x}^{\mathrm{2}} } {arctan}\left(\mathrm{2}{x}\right){dx} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
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Question Number 56311 by maxmathsup by imad last updated on 13/Mar/19 $${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({xt}\right)}{{x}^{\mathrm{2}} \:+{t}^{\mathrm{2}} }\:{dt}\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({t}\right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{cos}\left(\mathrm{2}{t}\right)}{\mathrm{4}+{t}^{\mathrm{2}}…
Question Number 56310 by maxmathsup by imad last updated on 13/Mar/19 $${let}\:{f}\left({x}\right)=\int_{−\infty} ^{+\infty} \:\:{cos}\left({t}^{\mathrm{2}} \:+{xt}\:+\mathrm{3}\right){dt}\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{1}} ^{\mathrm{4}} {f}\left({x}\right){dx}\:{and}\:\int_{\mathrm{1}} ^{+\infty} {f}\left({x}\right){dx} \\ $$…
Question Number 187374 by ajfour last updated on 16/Feb/23 $$\int\sqrt{\frac{{t}+\mathrm{1}}{{t}\left({k}−{t}\right)}}{dt}=? \\ $$ Commented by ajfour last updated on 16/Feb/23 $${see}\:{Q}.\mathrm{187364} \\ $$ Commented by MJS_new…
Question Number 121825 by bemath last updated on 12/Nov/20 $$\:\:\int\:\frac{{dx}}{{x}−\mathrm{4}\sqrt{{x}}}\:? \\ $$ Commented by bemath last updated on 12/Nov/20 $$\:\int\:\frac{{dx}}{\:\sqrt{{x}}\:\left(\sqrt{{x}}−\mathrm{4}\right)}\:{dx}\:=\:\mathrm{2}\int\:\frac{{d}\left(\sqrt{{x}}−\mathrm{4}\right)}{\:\sqrt{{x}}−\mathrm{4}} \\ $$$$=\:\mathrm{2}\:\mathrm{ln}\:\mid\sqrt{{x}}−\mathrm{4}\:\mid\:+\:{c}\: \\ $$ Answered…
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Question Number 56280 by rahul 19 last updated on 13/Mar/19 $${Evaluate}\:: \\ $$$$\left.\mathrm{1}\right)\:\frac{\int_{\mathrm{0}} ^{\:\mathrm{1}_{} } \left(\mathrm{1}−\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{100}} \right)^{\mathrm{201}} \:.{xdx}}{\int_{\mathrm{0}} ^{\:\mathrm{1}} \left(\mathrm{1}−\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{100}} \right)^{\mathrm{202}} .{xdx}}\:=\:? \\…
Question Number 187317 by ajfour last updated on 15/Feb/23 $$\int\frac{{dx}}{{x}\sqrt{\mathrm{1}−\mathrm{2}{x}}} \\ $$ Answered by MJS_new last updated on 15/Feb/23 $$\int\frac{{dx}}{{x}\sqrt{\mathrm{1}−\mathrm{2}{x}}}= \\ $$$$\:\:\:\:\:\left[{t}=\sqrt{\mathrm{1}−\mathrm{2}{x}}\:\rightarrow\:{dx}=−\sqrt{\mathrm{1}−\mathrm{2}{x}}{dt}\right] \\ $$$$=\mathrm{2}\int\frac{{dt}}{{t}^{\mathrm{2}} −\mathrm{1}}=\mathrm{ln}\:\frac{{t}−\mathrm{1}}{{t}+\mathrm{1}}\:=…
Question Number 121774 by mnjuly1970 last updated on 11/Nov/20 $$\:\:\:\:\:\:\:\:\:\:…\:{advanced}\:\:{calculus}… \\ $$$$\:\:\:\:\:\:\:{evaluate}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left({H}_{{n}} \right)^{\mathrm{2}} }{{n}^{\mathrm{2}} }\:=? \\ $$$$\:\:\:\:{where}\:\:\:{H}_{{n}} =\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{k}}\:\left({harmonic}\:{number}\right) \\…