What is the integral of ((ln(x^8)))/x ?

The integral of ((ln(x^8)))/x is 4ln^2(x)+C

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integral of ((ln(x^8)))/x

\int \frac{( ln ( x ^{8} ) )}{x} d x

4 ln^{2} (x) + C

Solution steps

\int \frac{ln ( x ^{8} )}{x} d x

Apply u-substitution

: \int \frac{ln ( u )}{8 u} d u

= \int \frac{ln ( u )}{8 u} d u

Take the constant out:

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

= \frac{1}{8} \cdot \int \frac{ln ( u )}{u} d u

Apply u-substitution

: \int v d v

= \frac{1}{8} \cdot \int v d v

Apply the Power Rule

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

= \frac{1}{8} \cdot \frac{v ^{2}}{2}

Substitute back

= \frac{1}{8} \cdot \frac{ln ^{2} ( x ^{8} )}{2}

Simplify

\frac{1}{8} \cdot \frac{ln ^{2} ( x ^{8} )}{2} : 4 ln^{2} (x)

= 4 ln^{2} (x)

Add a constant to the solution

= 4 ln^{2} (x) + C

\int sin (\frac{1}{3}) x d x

\frac{\partial}{\partial x} (9 y e^{x} - ln (z))

\frac{s + 2}{s ^{2} + 2 s + 5}

(t - 2) e^{- 4 t}

4 x + \frac{144}{x} + 80

What is the integral of ((ln(x^8)))/x ?
The integral of ((ln(x^8)))/x is 4ln^2(x)+C