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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (ln(x^2))/x

integral

\frac{ln ( x ^{2} )}{x}

ln^{2} (x) + C

Solution steps

\int \frac{ln ( x ^{2} )}{x} d x

Apply u-substitution

= \int \frac{ln ( u )}{2 u} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{ln ( u )}{u} d u

Apply u-substitution

= \frac{1}{2} \cdot \int v d v

Apply the Power Rule

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

Substitute back

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

Simplify

\frac{1}{2} \cdot \frac{ln ^{2} ( x ^{2} )}{2} : ln^{2} (x)

= ln^{2} (x)

Add a constant to the solution

= ln^{2} (x) + C