Category: Integration

Question Number 86611 by Ar Brandon last updated on 30/Mar/20 $${\int }_1^e\frac{\ln \mathrm{x}}{\mathrm{x}+1}\mathrm{dx}$$ Answered by Ar Brandon last updated on 30/Mar/20 Terms of…

Question Number 152147 by john_santu last updated on 26/Aug/21 Answered by Ar Brandon last updated on 26/Aug/21 $$I_n={\int }_{-1}^1(\sqrt{x^2+\frac{1}{n}}-\mid x\mid )dx$$$$\:\:\:\:=\mathrm{2}\int_{\mathrm{0}} ^{\mathrm{1}}…

Question Number 152142 by Ar Brandon last updated on 26/Aug/21 $${\int }_0^1\frac{x}{{(1-x+x^2)}^2}\ln \left(\ln \frac{1}{x}\right)dx=-\frac{\gamma }{3}-\frac{1}{3}\ln \frac{6\sqrt{3}}{\pi }+\frac{\pi \sqrt{3}}{27}(5\ln 2\pi -6\ln \mathrm{\Gamma }\left(\frac{1}{6}\right))$$ Answered by mindispower last updated on 28/Aug/21 $$\int_{\mathrm{0}}…

Question Number 152112 by peter frank last updated on 25/Aug/21 $$\mathrm{If}{\mathrm{I}}_{\mathrm{n}}=\int \frac{\cos \mathrm{nx}}{\cos \mathrm{x}}\mathrm{dx}\mathrm{then}{1}_{\mathrm{n}}=?$$ Answered by Olaf_Thorendsen last updated on 26/Aug/21 $$\mathrm{I}_{{n}} \:=\:\int\frac{\mathrm{cos}\left({nx}\right)}{\mathrm{cos}{x}}\:{dx} \…

Question Number 152116 by peter frank last updated on 25/Aug/21 $$\int \frac{\sin \mathrm{x}}{\sqrt{1+\sin \mathrm{x}}}\mathrm{dx}$$ Commented by puissant last updated on 26/Aug/21 $$Q=\int \frac{-cos(\frac{\pi }{2}+x)}{\sqrt{{(sin\left(\frac{x}{2}\right)+cos\left(\frac{x}{2}\right))}^2}}dx$$$$=−\int\frac{{cos}\left(\frac{\pi}{\mathrm{2}}+{x}\right)}{{sin}\left(\frac{{x}}{\mathrm{2}}\right)+{cos}\left(\frac{{x}}{\mathrm{2}}\right)}{dx} \…