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Evaluate the following definite integral...

Evaluate the following definite integral: `int_1^2e^(2x)(1/x-1/(2x^2))dx`

Text Solution

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`"Let I"=int_(1)^(2)((1)/(x)-(1)/(2x^(2)))e^(2x)dx`
`=int_(1)^(2)(1)/(x)e^(2x) dx- int_(1)^(2)(1)/(2x^(2))e^(2x)dx`
`=[(1)/(x).(e^(2x))/(2)]_(1)^(2)+int_(1)^(2)(1)/(x^(2)).(e^(2x))/(2)dx-int_(1)^(2)(1)/(2x^(2))e^(2x)dx`
`=[(1)/(x).(e^(2x))/(2)]_(1)^(2)=(e^(4))/(4)-(e^(2))/(2)=(e^(2))/(2)((e^(2))/(2)-1)`
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