Menu Close

Category: Integration

Question-51910

Question Number 51910 by Meritguide1234 last updated on 01/Jan/19 Commented by Abdo msup. last updated on 01/Jan/19 $${let}\:{find}\:{firstI}_{{n}} =\:\int_{\mathrm{0}} ^{\pi} \:{cos}^{{n}} {x}\:{cos}\left({nx}\right){dx}\:{with}\:{n}\:{from}\:{N} \\ $$$${I}_{{n}} ={Re}\left(\:\int_{\mathrm{0}}…

Question-51876

Question Number 51876 by Tinkutara last updated on 31/Dec/18 Commented by maxmathsup by imad last updated on 31/Dec/18 $${let}\:{solve}\:{y}^{'} \:+\mathrm{2}{y}\:={f}\left({x}\right)\:\:\left({e}\right) \\ $$$$\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}} \\…

0-1-arc-tan-x-2-dx-

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