What is the integral from 1 to e of (2ln|x|)/x ?

The integral from 1 to e of (2ln|x|)/x is 1

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integral from 1 to e of (2ln|x|)/x

\int_{1}^{e} \frac{2 ln ∣ x ∣}{x} d x

1

Solution steps

\int_{1}^{e} \frac{2 ln ∣ x ∣}{x} d x

Eliminate Absolutes

= \int_{1}^{e} \frac{2 ln ( x )}{x} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int_{1}^{e} \frac{ln ( x )}{x} d x

Apply u-substitution

: \int_{0}^{1} u d u

= 2 \cdot \int_{0}^{1} u d u

Apply the Power Rule

: [\frac{u ^{2}}{2}]_{0}^{1}

= 2 [\frac{u ^{2}}{2}]_{0}^{1}

Compute the boundaries:

\frac{1}{2}

= 2 \cdot \frac{1}{2}

Simplify

= 1

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What is the integral from 1 to e of (2ln|x|)/x ?
The integral from 1 to e of (2ln|x|)/x is 1