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

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

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

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

= \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

: \int v d v

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

Apply the Power Rule

: \frac{v ^{2}}{2}

= \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

\int \frac{1}{( x + 2 ) ^{5}} d x

\int e^{- x^{2}} x^{2} d x

cos (3 x)

\int \frac{3 x - 1}{x ^{3} + x} d x

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

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