Category: Integration

Question Number 42773 by maxmathsup by imad last updated on 02/Sep/18 $$letf\left(x\right)={\int }_0^1\frac{e^t}{1+x^t}dtwith0<x<1$$$$givef\left(x\right)atformofserie.$$ Terms of Service Privacy Policy…

Question Number 108306 by Sayantan chakraborty last updated on 16/Aug/20 $$\mathrm{proof}\mathrm{that}\underset{0}{\stackrel{\pi /4}{\int }}\frac{{sin}^2{xcos}^{2}x}{{({sin}^{3}x+{cos}^{3}x)}^2}dx=\frac{1}{6}$$ Terms of Service Privacy Policy…

Question Number 42769 by maxmathsup by imad last updated on 02/Sep/18 $$find{\int }_0^1\frac{x^2}{1+{xe}^{-x}}dx.$$ Terms of Service Privacy Policy Contact: info@tinkutara.com

Question Number 108303 by Sayantan chakraborty last updated on 16/Aug/20 $$\underset{0}{\stackrel{\pi /4}{\int }}\frac{1}{(sinx+cosx)}{sin}^{6}xdxequals$$ Answered by Sayantan chakraborty last updated on 16/Aug/20 $$\mathrm{Answer}:\frac{\mathrm{1}}{\mathrm{4}}\left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\mathrm{ln}\left(\sqrt{}\mathrm{2}+\mathrm{1}\right)+\mathrm{1}\right).…

Question Number 108283 by Ar Brandon last updated on 16/Aug/20 $$\mathrm{Given}\mathrm{the}\mathrm{function}\mathrm{\Gamma }\mathrm{defined}\mathrm{by}\mathrm{\Gamma }\left(\mathrm{x}\right)={\int }_0^{+\mathrm{\infty }}{\mathrm{t}}^{\mathrm{x}-1}{\mathrm{e}}^{-\mathrm{t}}\mathrm{dt}$$$$1.\mathrm{What}\mathrm{is}\mathrm{the}\mathrm{domain}\mathrm{of}\mathrm{definition}\mathrm{of}\mathrm{\Gamma }?$$$$2.\mathrm{Show}\mathrm{that}\mathrm{\forall }\mathrm{x}\in \mathrm{D}\mathrm{\Gamma },\mathrm{x}\mathrm{\Gamma }\left(\mathrm{x}\right)=\mathrm{\Gamma }(\mathrm{x}+1)\mathrm{and}\mathrm{deduce}\mathrm{the}\mathrm{value}\mathrm{of}\mathrm{\Gamma }\left(\mathrm{n}\right),\mathrm{n}\in {\mathbb{N}}^{\ast }$$$$\mathrm{3}.\:\:\mathrm{Assuming}\:\int_{\mathrm{0}} ^{+\infty} \mathrm{e}^{−\mathrm{u}^{\mathrm{2}} } =\frac{\sqrt{\pi}}{\mathrm{2}},\:\mathrm{calculate}\:\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\:\mathrm{and}\:\mathrm{deduce}\:\mathrm{that}…