Category: Integration

Question Number 172635 by Mikenice last updated on 29/Jun/22 Answered by MJS_new last updated on 29/Jun/22 $$\int \frac{4^x}{4^x+1}dx=$$$$[t=4^x+1\to dx=\frac{dt}{4^x\ln 4}]$$$$=\frac{\mathrm{1}}{\mathrm{ln}\:\mathrm{4}}\int\frac{{dt}}{{t}}=\frac{\mathrm{ln}\:{t}}{\mathrm{ln}\:\mathrm{4}}=…

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Question Number 41518 by maxmathsup by imad last updated on 08/Aug/18 $$calculate{A}_n={\int }_0^1{(1-t^2)}^{n}dtwithnintegrnatural$$ Answered by alex041103 last updated on…

Question Number 41515 by maxmathsup by imad last updated on 08/Aug/18 $$let{f}_n\left(x\right)=\frac{sin\left(2(n+1)x\right)}{sinx}ifx\in ]0,\frac{\pi }{2}]and{f}_n\left(0\right)=2(n+1)let$$$$u_n={\int }_0^{\frac{\pi }{2}}{f}_n\left(x\right)dx$$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\forall{n}\:{fromN}\:\:{u}_{{n}+\mathrm{1}} −{u}_{{n}} =\mathrm{2}\frac{\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} }{\mathrm{2}{n}+\mathrm{3}}…