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

h (t) = (t + 5)^{\frac{2}{3}} (3 t^{2} - 1)^{3}

y = sin (4 x)

\int (cos (x))^{3} sin (x) d x

\int_{0}^{3} ((30 - y) - y^{\frac{1}{3}}) d y

\frac{d}{d x} (sin (x) - e^{x})

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