Question Number 151983 by puissant last updated on 24/Aug/21 $$\int_{−\infty} ^{+\infty} \left(\frac{\mathrm{1}−{ix}}{\mathrm{1}+{ix}}\right)^{{n}} \left(\frac{\mathrm{1}+{ix}}{\mathrm{1}−{ix}}\right)^{{m}} \frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{2}} }{dx} \\ $$ Answered by Olaf_Thorendsen last updated on 24/Aug/21 $$\mathrm{I}\:=\:\int_{−\infty}…
Question Number 20905 by NECx last updated on 07/Sep/17 $$\int{e}^{{x}^{\mathrm{2}} } {dx} \\ $$ Commented by NECx last updated on 07/Sep/17 $${please}\:{help} \\ $$ Answered…
Question Number 86431 by M±th+et£s last updated on 28/Mar/20 $$\int\sqrt{{x}−\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}\:}\:{dx} \\ $$ Answered by jagoll last updated on 28/Mar/20 $$\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{1}}\:=\:\mathrm{x}−\mathrm{t}\: \\ $$$$\mathrm{x}^{\mathrm{2}} +\mathrm{1}\:=\:\mathrm{x}^{\mathrm{2}}…
Question Number 86428 by jagoll last updated on 28/Mar/20 $$\int\:\:\frac{\mathrm{dx}}{\mathrm{a}\:\mathrm{cos}\:\mathrm{x}\:+\:\mathrm{b}\:\mathrm{sin}\:\mathrm{x}}? \\ $$ Commented by jagoll last updated on 28/Mar/20 $$\mathrm{standard}\:\mathrm{solving} \\ $$$$\mathrm{a}\:\mathrm{cos}\:\mathrm{x}\:+\:\mathrm{b}\:\mathrm{sin}\:\mathrm{x}\:=\:\mathrm{k}\:\mathrm{cos}\:\left(\mathrm{x}−\theta\right) \\ $$$$\int\:\frac{\mathrm{dx}}{\mathrm{k}\:\mathrm{cos}\:\left(\mathrm{x}−\theta\right)}\:=\:\frac{\mathrm{1}}{\mathrm{k}}\int\:\mathrm{sec}\:\left(\mathrm{x}−\theta\right)\:\mathrm{dx} \\…
Question Number 151958 by mnjuly1970 last updated on 24/Aug/21 $${f}\:\left(\:{x}\:\right)\:=\:{a}\:−\sqrt{\frac{{x}}{\mathrm{1}+{x}}\:}\:\:\:,\:{D}_{\:{f}} \::\:\left[\:\mathrm{0},\:\infty\right) \\ $$$$,\:{a}\geqslant\:\mathrm{1}\:\:\:,\:\:{h}\:\left({x}\:\right):=\sqrt{\frac{\:{f}^{\:−\mathrm{1}} \left({a}−{ax}\:\right)}{{f}^{\:−\mathrm{1}} \left(\:{a}−\:\mathrm{2}{x}\:\right)}} \\ $$$$\:\:\:\:\:\:\:\:\:\:{D}_{\:{h}} \:=\:?\:\:\:\left(\:\:\:{D}\::=\:{Domain}\:\right) \\ $$$$\:\:\:\:\:\:\:\:\:\: \\ $$ Terms of Service…
Question Number 151951 by mnjuly1970 last updated on 24/Aug/21 $$ \\ $$$$\:\:\:\:\:{nice}\:…\:{mathematics} \\ $$$$\:\:\:\:\:\:\mathrm{S}:=\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:\zeta\:\left(\mathrm{2}{n}\:\right)}{{n}\:.\:\mathrm{16}^{\:{n}} }\:=\:?\:……\blacksquare \\ $$$$ \\ $$ Commented by Tawa11 last…
Question Number 20873 by Joel577 last updated on 05/Sep/17 $$\int_{\mathrm{1}} ^{\mathrm{5}} \frac{{e}^{{x}} }{{x}^{\mathrm{2}} }\:{dx} \\ $$ Answered by alex041103 last updated on 08/Sep/17 $${First}\:{we}\:{apply}\:{integration}\:{by}\:{parts} \\…
Question Number 86407 by Power last updated on 28/Mar/20 Commented by mathmax by abdo last updated on 28/Mar/20 $$\left.{A}\:=\int\:\sqrt{\left({x}−{a}\right)\left({b}−{x}\right.}\right){dx}\:\Rightarrow{A}\:=\int\sqrt{{x}−{a}}\sqrt{{b}−{x}}{dx}\:{chsngement} \\ $$$$\sqrt{{x}−{a}}={t}\:{give}\:{x}−{a}\:={t}^{\mathrm{2}} \:\Rightarrow \\ $$$${A}\:=\int\:\:{t}\sqrt{{b}−\left({a}+{t}^{\mathrm{2}} \right)}\left(\mathrm{2}{t}\right){dt}\:=\mathrm{2}\:\int\:{t}^{\mathrm{2}}…
Question Number 86405 by Power last updated on 28/Mar/20 Commented by MJS last updated on 28/Mar/20 $$\mathrm{sin}^{\mathrm{2}} \:\mathrm{3}{x}\:\mathrm{sin}^{\mathrm{3}} \:\mathrm{2}{x}\:= \\ $$$$=\frac{\mathrm{1}}{\mathrm{16}}\left(\mathrm{sin}\:\mathrm{12}{x}\:−\mathrm{3sin}\:\mathrm{8}{x}\:−\mathrm{2sin}\:\mathrm{6}{x}\:+\mathrm{3sin}\:\mathrm{4}{x}\:+\mathrm{6sin}\:\mathrm{2}{x}\right) \\ $$$$\mathrm{now}\:\mathrm{it}'\mathrm{s}\:\mathrm{super}\:\mathrm{easy} \\ $$…
Question Number 86399 by Power last updated on 28/Mar/20 Commented by Prithwish Sen 1 last updated on 28/Mar/20 $$\mathrm{put}\:\mathrm{x}=\mathrm{2asin}^{\mathrm{2}} \mathrm{t} \\ $$$$\mathrm{and}\:\mathrm{simplify}. \\ $$ Commented…