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
Questions 203525
Subjects 9
Chapters 126
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