Category: Integration

Question Number 146791 by puissant last updated on 15/Jul/21 $$\int \frac{\mathrm{x}}{1+{\cos }^2\left(\mathrm{x}\right)}\mathrm{dx}$$ Answered by Ar Brandon last updated on 15/Jul/21 $$\mathrm{I}=\int\frac{\mathrm{x}}{\mathrm{1}+\mathrm{cos}^{\mathrm{2}} \mathrm{x}}\mathrm{dx}=\int\frac{\mathrm{xsec}^{\mathrm{2}} \mathrm{x}}{\mathrm{sec}^{\mathrm{2}} \mathrm{x}+\mathrm{1}}\mathrm{dx}…

Question Number 146756 by Bens last updated on 15/Jul/21 $$$$$$$$$$\underset{0}{\stackrel{1}{\int }}{t}^2+1dt$$ Answered by tabata last updated on…

Question Number 146736 by mnjuly1970 last updated on 15/Jul/21 $$$$$$1:\mathrm{S}:=\underset{n=1}{\sum ^{\mathrm{\infty }}}\frac{{(-1)}^{n-1}}{n.2^n}=?$$$$2:\mathrm{A}:=\mathrm{\Sigma }\frac{{(-1)}^{n-1}}{n^2.2^n}=?$$ Answered by…

Question Number 146735 by qaz last updated on 15/Jul/21 $$\underset{\mathrm{n}=1}{\sum ^{\mathrm{\infty }}}\frac{1+\frac{1}{2}+\frac{1}{3}+\dots +\frac{1}{\mathrm{n}}}{(\mathrm{n}+1)(\mathrm{n}+2)}=?$$ Answered by Olaf_Thorendsen last updated on 15/Jul/21 $$\mathrm{S}_{\mathrm{N}} \:=\:\underset{{n}=\mathrm{1}} {\overset{\mathrm{N}} {\sum}}\frac{{H}_{{n}}…

Question Number 146697 by ArielVyny last updated on 15/Jul/21 $$\mathrm{\forall }t⩾-1,F\left(t\right)=\frac{2}{\pi }{\int }_0^{\frac{\pi }{2}}\sqrt{1+{tcos}^2\phi }d\phi$$$$1)Showthat\mathrm{\forall }t⩽-1F\left(t\right)=\sqrt{1+t}F(-\frac{1}{1+t})$$$$2)showthatif0⩽t_1,$$$$\mathrm{0}\leqslant{F}\left({t}_{\mathrm{2}} \right)−{F}\left({t}_{\mathrm{1}} \right)\leqslant\frac{{t}_{\mathrm{2}} −{t}_{\mathrm{1}} }{\mathrm{4}} \…