Question

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

Answer 1

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}`