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Evaluate \int _{1}^{2}\left [ \frac{1}{x}-\frac{1}{2x^{2}} \right ]e^{2x}dx

 

 

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\int _{1}^{2}\left [ \frac{1}{x}-\frac{1}{2x^{2}} \right ]e^{2x}dx

\\\text { Put } 2 x=t \\ d x=\frac{1}{2} d t \\\\ I=\int_{1}^{2}\left[\frac{1}{x}-\frac{1}{2 x^{2}}\right] e^{2 x} d x \\\\ =\int_{2}^{4}\left[\frac{1}{t}-\frac{1}{t^{2}}\right] e^{t} d t \\\\ =\left[\frac{1}{t} e^{t}\right]_{2}^{4} \because \int (f(x)+f'(x))e^xdx = e^x +c \\\\ =\frac{e^{4}}{4}-\frac{e^{2}}{2}

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