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

\frac{1}{3 x - 1}

\frac{d y}{d x} + y = x, y (0) = 4

\int \frac{10 x}{2 + 5 x ^{2}} d x

x \to - 1 lim (\frac{x + 1}{x - 1})

y = x^{3} + 1, (1, 2)