Category: Integration

Question Number 51876 by Tinkutara last updated on 31/Dec/18 Commented by maxmathsup by imad last updated on 31/Dec/18 $$letsolve{y}^{{}^{\prime }}+2y=f\left(x\right)\left(e\right)$$$$\left({he}\right)\:\Rightarrow{y}^{'} \:+\mathrm{2}{y}\:=\mathrm{0}\:\Rightarrow\frac{{y}^{'} }{{y}}=−\mathrm{2}\:\:\Rightarrow{ln}\left({y}\right)=−\mathrm{2}{x}\:+{k}\:\:\Rightarrow{y}\left({x}\right)={C}\:{e}^{−\mathrm{2}{x}} \…

Question Number 117403 by bemath last updated on 11/Oct/20 $$\underset{0}{\stackrel{1}{\int }}{\left(\mathrm{arc}\tan \mathrm{x}\right)}^2\mathrm{dx}=?$$ Commented by MJS_new last updated on 11/Oct/20 $$\mathrm{use}\arctan x=\frac{\ln (1+\mathrm{i}x)-\ln (1-\mathrm{i}x)}{2\mathrm{i}}\Rightarrow$$$$−\frac{\mathrm{1}}{\mathrm{4}}\underset{\mathrm{0}}…

Question Number 117380 by mnjuly1970 last updated on 11/Oct/20 $$\dots provethat\dots$$$$$$$$\mathrm{\Omega }={\int }_0^{\mathrm{\infty }}\frac{1}{2\sqrt{x}}sin({\pi }^{2}x+\frac{1}{x})dx=\frac{1}{\sqrt{8\pi }}$$$$$$$$m.n.1970$$ Commented by…

Question Number 51834 by Abdo msup. last updated on 31/Dec/18 $$calculatef\left(a\right)={\int }_0^{\mathrm{\infty }}\frac{ln(1+{at}^2)}{1+t^4}dtwitha>0.$$$$2)findthevalueof{\int }_0^{\mathrm{\infty }}\frac{ln(3+t^2)}{1+t^4}dt.$$ Commented by…

Question Number 51825 by Abdo msup. last updated on 30/Dec/18 $$findf\left(\alpha \right)={\int }_0^{1}ln(1+e^{-\alpha }x)dxwith\alpha ⩾0$$ Commented by maxmathsup by imad last updated on 31/Dec/18…