What is the integral of (2ln|x|)/x ?

The integral of (2ln|x|)/x is ln|x|^2+C

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integral of (2ln|x|)/x

\int \frac{2 ln ∣ x ∣}{x} d x

ln ∣ x ∣^{2} + C

Solution steps

\int \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 \frac{ln ∣ x ∣}{x} d x

\frac{ln ∣ x ∣}{x} = ln ∣ x ∣ \frac{1}{x}

= 2 \cdot \int ln ∣ x ∣ \frac{1}{x} d x

Apply u-substitution

= 2 \cdot \int u d u

Apply the Power Rule

= 2 \cdot \frac{u ^{2}}{2}

Substitute back

u = ln ∣ x ∣

= 2 \cdot \frac{ln ∣ x ∣ ^{2}}{2}

Simplify

2 \cdot \frac{ln ∣ x ∣ ^{2}}{2} : ln ∣ x ∣^{2}

= ln ∣ x ∣^{2}

Add a constant to the solution

= ln ∣ x ∣^{2} + C