How do you find the integral $int_1^2e^(1/x)/x^2dx$ ?

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Answer 1 · 2014-09-11T23:41:13

By using the substitution $u=1/x$ , we can find
$int_1^2e^{1/x}/x^2dx=e-sqrt{e}$ .

Let $u=1/x$ . $Rightarrow {du}/{dx}=-1/x^2 Rightarrow dx=-x^2du$
Since $x$ goes from 1 to 2, $u$ goes from 1 to 1/2.

Let us look at some details,
$int_1^2e^{1/x}/x^2dx$

by Substitution,
$=int_1^{1/2}e^u/x^2cdot(-x^2)du$

by cancelling $x^2$ ,
$=-int_1^{1/2}e^udu$

by using the negative sign to switch the lower and upper limits,
$=int_{1/2}^1e^udu$

by taking the antiderivative,
$=[e^u]_{1/2}^1=e-e^{1/2}=e-sqrt{e}$

How do you find the integral int_1^2e^(1/x)/x^2dxint_1^2e^(1/x)/x^2dx ?