Question Number 216742 by Tawa11 last updated on 17/Feb/25 $$\:\frac{\mathrm{1}}{\mathrm{2}}\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{ln}\left(\mathrm{a}\:\:−\:\:\mathrm{1}\right)}{\mathrm{a}}\:\mathrm{da} \\ $$ Answered by sniper237 last updated on 17/Feb/25 $${Not}\:{defined}\:! \\ $$$${But}\:\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\mathrm{1}}…
Question Number 216715 by Nadirhashim last updated on 16/Feb/25 $$\:\:\:\:\boldsymbol{{F}}{ind}\:\int\frac{\boldsymbol{{S}}{in}\left(\frac{\mathrm{5}{x}\:}{\mathrm{2}\:}\right)\:\:}{\boldsymbol{{S}}{in}\left(\frac{{x}\:}{\mathrm{2}\:}\right)\:\:\:\:\:}\:.\boldsymbol{{dx}}\:\:\: \\ $$ Answered by Frix last updated on 16/Feb/25 $$\int\frac{\mathrm{sin}\:\frac{\mathrm{5}{x}}{\mathrm{2}}}{\mathrm{sin}\:\frac{{x}}{\mathrm{2}}}{dx}=\int\left(\mathrm{2cos}\:\mathrm{2}{x}\:+\mathrm{2cos}\:{x}\:+\mathrm{1}\right){dx}= \\ $$$$=\mathrm{sin}\:\mathrm{2}{x}\:+\mathrm{2sin}\:{x}\:+{x}+{C} \\ $$ Answered…
Question Number 216618 by Tawa11 last updated on 12/Feb/25 $$\int_{\:\mathrm{0}} ^{\:\mathrm{2}\pi} \:\sqrt{\mathrm{1}\:\:−\:\:\mathrm{cos}^{\mathrm{2}} \mathrm{x}}\:\:\mathrm{dx} \\ $$$$\mathrm{Is}\:\mathrm{the}\:\mathrm{answer}\:\:\mathrm{0}\:\:\:\mathrm{or}\:\:\:\:\mathrm{4}???? \\ $$ Answered by A5T last updated on 12/Feb/25 $$\int_{\mathrm{0}}…
Question Number 216613 by ajfour last updated on 12/Feb/25 $$\int_{{a}} ^{\:{x}} \frac{\mathrm{1}−{b}\mathrm{ln}\:\frac{{x}}{{a}}}{\:\sqrt{\mathrm{1}−\left(\mathrm{1}−{b}\mathrm{ln}\:\frac{{x}}{{a}}\right)^{\mathrm{2}} }}\:{dx} \\ $$ Commented by ajfour last updated on 12/Feb/25 https://youtu.be/Kq4VqD0azog?si=K_P6MADAgm8CXegJ Terms of…
Question Number 216579 by MrGaster last updated on 11/Feb/25 $$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \frac{\mathrm{tan}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)+\mathrm{sin}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)}{\mathrm{tan}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)+\mathrm{cos}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)}{dxdy} \\ $$ Terms…
Question Number 216491 by Jubr last updated on 09/Feb/25 Answered by MrGaster last updated on 09/Feb/25 $$\left(\mathrm{1}\right): \\ $$$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{x}\int_{\mathrm{0}} ^{{x}} \left(\mathrm{1}−{t}^{\mathrm{2}} +\frac{{t}^{\mathrm{4}} }{\mathrm{2}!}−\frac{{t}^{\mathrm{6}} }{\mathrm{3}!}+\ldots\right){dt}}{\:\sqrt{\mathrm{1}−{e}^{−{x}^{\mathrm{2}}…
Question Number 216486 by Tawa11 last updated on 08/Feb/25 $$\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\mathrm{x}\sqrt{\mathrm{x}\:\:\sqrt[{\mathrm{3}}]{\mathrm{x}\:\:\sqrt[{\mathrm{4}}]{\mathrm{x}\:\:\sqrt[{\mathrm{5}}]{\mathrm{x}\:…}}}}\:\:\mathrm{dx} \\ $$ Answered by mehdee7396 last updated on 09/Feb/25 $${x}×{x}^{\frac{\mathrm{1}}{\mathrm{2}}} ×{x}^{\frac{\mathrm{1}}{\mathrm{6}}} ×{x}^{\frac{\mathrm{1}}{\mathrm{24}}} ×{x}^{\frac{\mathrm{1}}{\mathrm{120}}}…
Question Number 216408 by MrGaster last updated on 07/Feb/25 $$\int_{−\mathrm{1}} ^{\mathrm{1}} \frac{\mathrm{1}}{{x}}\sqrt{\frac{\mathrm{1}+{x}}{\mathrm{1}−{x}}}\mathrm{ln}\left(\frac{\mathrm{2}{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{1}}{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{1}}\right){dx} \\ $$ Commented by MrGaster last updated on 07/Feb/25 $${I}=\int_{−\mathrm{1}} ^{+\mathrm{1}}…
Question Number 216381 by glory86 last updated on 06/Feb/25 $$\int\left({lnx}\right)^{\mathrm{2}} {dx} \\ $$ Answered by mr W last updated on 06/Feb/25 $$\int\left(\mathrm{ln}\:{x}\right)^{\mathrm{2}} {dx} \\ $$$$={x}\left(\mathrm{ln}\:{x}\right)^{\mathrm{2}}…
Question Number 216372 by glory86 last updated on 06/Feb/25 $$\int\frac{{xe}^{{x}} }{\left({x}+\mathrm{1}\right)^{\mathrm{2}} }{dx} \\ $$ Answered by MrGaster last updated on 06/Feb/25 $$\mathrm{Let}\:{u}={x}+\mathrm{1}\Rightarrow{du}={dx},{x}={u}−\mathrm{1} \\ $$$$\int\frac{\left({u}−\mathrm{1}\right){e}^{{u}−\mathrm{1}} }{{u}^{\mathrm{2}}…