What is the integral from 1 to e of |ln(x)| ?

The integral from 1 to e of |ln(x)| is 1

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integral from 1 to e of |ln(x)|

\int_{1}^{e} ∣ ln (x) ∣ d x

1

Solution steps

\int_{1}^{e} ∣ ln (x) ∣ d x

Apply u-substitution

= \int_{0}^{1} ∣ u ∣ e^{u} d u

Eliminate Absolutes

= \int_{0}^{1} u e^{u} d u

Apply Integration By Parts

= [e^{u} u - \int e^{u} d u]_{0}^{1}

\int e^{u} d u = e^{u}

= [e^{u} u - e^{u}]_{0}^{1}

Compute the boundaries:

1

= 1

What is the integral from 1 to e of |ln(x)| ?
The integral from 1 to e of |ln(x)| is 1