Mathematics

Evaluate the following definite integral:

$$\displaystyle \int_{1}^{2} e^{2x} \left (\dfrac {1}{x}-\dfrac {1}{2x^2}\right)dx$$.


SOLUTION
$$I=\displaystyle \int_{1}^{2} e^{2x} \left (\dfrac {1}{x}-\dfrac {1}{2x^2} \right)dx =\displaystyle \int_{1}^{2} \begin{matrix} { e }^{ 2x } \\ \begin{matrix}  \\ \end{matrix} \end{matrix}\frac { 1 }{ \begin{matrix} x \\  \end{matrix} } dx -\displaystyle \int_{1}^{2} e^{2x} \dfrac {1}{2x^2}dx$$

$$\Rightarrow \ I=\left [\dfrac {1}{2x}e^{2x}\right]_1^2 +\displaystyle \int_{1}^{2} \dfrac {1}{2x^2}e^{2x}dx-\displaystyle \int_1^2 e^{2x} \dfrac {1}{2x^2}dx=\left (\dfrac {1}{4} e^4 -\dfrac {1}{2}e^2 \right) =\dfrac {e^4 -2e^2}{4}$$

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Subjective Medium Published on 17th 09, 2020
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