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Question Number 54699 by rahul 19 last updated on 09/Feb/19

$$\int e^{2x}(\frac{1}{x}-\frac{1}{2x^2})dx$$$$\int e^{2x}\left(\frac{1+\sin 2x}{1+\cos 2x}\right)dx.$$$$SolveaboveQuestionsbyusingthe$$$$formulae:$$$$\int e^{kx}\{f\left(kx\right)+f{}^{\prime }\left(kx\right)\}dx=e^{kx}f\left(kx\right)+c.$$

Commented by maxmathsup by imad last updated on 10/Feb/19

$$letI=\int (\frac{1}{x}-\frac{1}{2x^2})e^{2x}dx\Rightarrow I=\int \frac{e^{2x}}{x}dx-\int \frac{e^{2x}}{2x^2}bypsrts$$$$\int \frac{e^{2x}}{x}dx=\frac{1}{2x}{e}^{2x}-\int -\frac{1}{2x^2}{e}^{2x}=\frac{1}{2x}{e}^{2x}+\int \frac{e^{2x}}{2x^2}\Rightarrow$$$$I=\frac{e^{2x}}{2x}+\int \frac{e^{2x}}{2x^2}-\int \frac{e^{2x}}{2x^2}dx+c\Rightarrow I=\frac{e^{2x}}{2x}+c.$$

Commented by rahul 19 last updated on 10/Feb/19

thank you profAbdo .

Commented by maxmathsup by imad last updated on 11/Feb/19

$$youarewelcomesir.$$

Answered by kaivan.ahmadi last updated on 09/Feb/19

$$\mathrm{f}\left(\mathrm{x}\right)=\frac{2}{\mathrm{x}}\Rightarrow \mathrm{f}\left(2\mathrm{x}\right)=\frac{1}{\mathrm{x}}\Rightarrow {\mathrm{f}}^{\prime }\left(\mathrm{x}\right)=\frac{-2}{{\mathrm{x}}^2}\Rightarrow$$$${\mathrm{f}}^{\prime }\left(2\mathrm{x}\right)=\frac{-1}{{2\mathrm{x}}^2}$$$$\int {\mathrm{e}}^{2\mathrm{x}}(\frac{1}{\mathrm{x}}-\frac{1}{{2\mathrm{x}}^2})\mathrm{dx}={\mathrm{e}}^{2\mathrm{x}}\mathrm{f}\left(2\mathrm{x}\right)+\mathrm{c}=\frac{{\mathrm{e}}^{2\mathrm{x}}}{\mathrm{x}}+\mathrm{c}$$$$$$$$$$$$$$

Commented by rahul 19 last updated on 09/Feb/19

$$Sir,iff\left(2x\right)=\frac{1}{x}thenf{}^{\prime }\left(2x\right)shouldbe$$$$\frac{-1}{x^2}?{Isn}^{\prime }tit?$$

Commented by kaivan.ahmadi last updated on 09/Feb/19

$$\mathrm{yes}\mathrm{it}\mathrm{is},\mathrm{thank}\mathrm{you}.\mathrm{the}\mathrm{answer}\mathrm{of}\mathrm{your}\mathrm{integral}\mathrm{is}\mathrm{not}\mathrm{true},\mathrm{because}$$$${\left({\mathrm{e}}^{\mathrm{kx}}\mathrm{f}\left(\mathrm{kx}\right)\right)}^{{}^{\prime }}={\mathrm{ke}}^{2\mathrm{x}}(\mathrm{f}\left(\mathrm{kx}\right)+{\mathrm{f}}^{\prime }\left(\mathrm{kx}\right))$$$$$$$$$$$$$$

Answered by tanmay.chaudhury50@gmail.com last updated on 09/Feb/19

$$1)\frac{1}{2}\int [\frac{1}{x}\times \frac{d}{dx}\left(e^{2x}\right)+e^{2x}\frac{d}{dx}\left(\frac{1}{x}\right)]dx$$$$\frac{1}{2}\int \frac{d}{dx}\left(\frac{1}{x}{e}^{2x}\right)dx$$$$\frac{1}{2}\left(\frac{e^{2x}}{x}\right)+c$$$$2)\int e^{2x}\left(\frac{1+2sinxcosx}{2{cos}^{2}x}\right)$$$$\int e^{2x}(\frac{1}{2}{sec}^{2}x+tanx)dx$$$$\frac{1}{2}\int [e^{2x}\times \frac{d}{dx}\left(tanx\right)+tanx\times \frac{d}{dx}\left(e^{2x}\right)]dx$$$$=\frac{1}{2}\int \frac{d}{dx}\left(e^{2x}tanx\right)dx$$$$\frac{e^{2x}tanx}{2}+c$$

Commented by rahul 19 last updated on 10/Feb/19

thank you sir �� but how to do by using given formulae ?

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