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Question Number 502 by 123456 last updated on 25/Jan/15 $$\underset{−\mathrm{1}} {\overset{+\mathrm{1}} {\int}}\frac{\mathrm{ln}\:\mid{x}\mid}{{x}^{\mathrm{2}} −\mathrm{1}}{dx} \\ $$ Answered by prakash jain last updated on 20/Jan/15 $${I}=\underset{−\mathrm{1}} {\overset{+\mathrm{1}}…
Question Number 499 by 123456 last updated on 18/Jan/15 $$\underset{−\infty} {\overset{\mathrm{0}} {\int}}\:\frac{{e}^{{x}} \mathrm{ln}\:\left(−{x}\right)}{{x}}{dx} \\ $$ Commented by prakash jain last updated on 19/Jan/15 $$\mathrm{Doesn}'\mathrm{t}\:\mathrm{converge}. \\…
Question Number 492 by 123456 last updated on 25/Jan/15 $$\int\frac{\mathrm{1}−\sqrt{{x}}}{\mathrm{1}+\sqrt{{x}}}{dx} \\ $$ Answered by prakash jain last updated on 14/Jan/15 $${x}={t}^{\mathrm{2}} \\ $$$${dx}=\mathrm{2}{t}\:{dt} \\ $$$$\int\frac{\mathrm{1}−{t}}{\mathrm{1}+{t}}\centerdot\mathrm{2}{t}\:{dt}=\mathrm{2}\int\:\frac{{t}−{t}^{\mathrm{2}}…
Question Number 131558 by liberty last updated on 06/Feb/21 $$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{dx}}{\:\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{4}} }}\:=? \\ $$ Answered by Dwaipayan Shikari last updated on 06/Feb/21 $$\int_{\mathrm{0}} ^{\mathrm{1}}…
Question Number 131552 by liberty last updated on 06/Feb/21 $$\int\:\frac{\mathrm{x}\:\mathrm{dx}}{\left(\mathrm{cot}\:\mathrm{x}+\mathrm{tan}\:\mathrm{x}\right)^{\mathrm{2}} }\:? \\ $$ Answered by EDWIN88 last updated on 06/Feb/21 $$\Leftrightarrow\:\mathrm{cot}\:\mathrm{x}+\mathrm{tan}\:\mathrm{x}\:=\:\frac{\mathrm{1}}{\mathrm{sin}\:\mathrm{x}\:\mathrm{cos}\:\mathrm{x}}\:=\:\frac{\mathrm{2}}{\mathrm{sin}\:\mathrm{2x}} \\ $$$$\Leftrightarrow\:\frac{\mathrm{1}}{\left(\mathrm{cot}\:\mathrm{x}+\mathrm{tan}\:\mathrm{x}\right)^{\mathrm{2}} }=\frac{\mathrm{sin}\:^{\mathrm{2}} \mathrm{2x}}{\mathrm{4}}…
Question Number 131549 by liberty last updated on 06/Feb/21 $$\mathrm{slowly}\:\mathrm{integral}\: \\ $$$$\int\:\frac{\mathrm{sec}\:^{\mathrm{4}} \mathrm{x}}{\:\sqrt{\mathrm{tan}\:^{\mathrm{3}} \mathrm{x}}}\:\mathrm{dx}\:=? \\ $$ Answered by rs4089 last updated on 06/Feb/21 $$\frac{\mathrm{2}{tan}^{\mathrm{2}} {x}−\mathrm{6}}{\mathrm{3}\sqrt{{tanx}}}+{c}\:\:\:\:\left\{{c}\:{is}\:{a}\:{constant}\right\}…
Question Number 131546 by mathmax by abdo last updated on 05/Feb/21 $$\mathrm{prove}\:\mathrm{that}\:\int_{\mathrm{0}} ^{\infty} \:\mathrm{e}^{−\mathrm{x}} \mathrm{ln}\left(\mathrm{x}\right)\mathrm{dx}\:=−\gamma \\ $$ Answered by Dwaipayan Shikari last updated on 06/Feb/21…
Question Number 131547 by liberty last updated on 06/Feb/21 $$\:\mathrm{J}\:=\:\int_{\mathrm{0}} ^{\:\infty} \:\frac{\mathrm{x}^{\mathrm{8}} −\mathrm{1}}{\mathrm{x}^{\mathrm{10}} +\mathrm{1}}\:\mathrm{dx}\:? \\ $$ Answered by rs4089 last updated on 06/Feb/21 $$\mathrm{0} \\…
Question Number 131529 by mnjuly1970 last updated on 05/Feb/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…{advanced}\:\:\:\ast\ast\ast\ast\ast\ast\ast\ast\ast\ast\:\:\:\:{calculus}… \\ $$$$\:\:\:{prove}\:\:{that}\::::\::: \\ $$$$\:\:\:\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}\left({x}^{\mathrm{4}} \right){ln}\left({x}\right)}{{x}}{dx}=−\frac{\boldsymbol{\pi\gamma}}{\mathrm{32}} \\ $$$$\:\:\:\:{note}\::\:\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}\left({x}\right){ln}\left({x}\right)}{{x}}{dx}\overset{{why}???} {=}\:\frac{−\boldsymbol{\pi\gamma}}{\mathrm{2}} \\ $$$$\:\:\:\:\boldsymbol{\phi}\overset{\langle{x}^{\mathrm{4}} ={t}\rangle}…