What is the integral of (ln(x^{24}))/x ?

The integral of (ln(x^{24}))/x is 12ln^2(x)+C

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integral of (ln(x^{24}))/x

\int \frac{ln ( x ^{24} )}{x} d x

12 ln^{2} (x) + C

Solution steps

\int \frac{ln ( x ^{24} )}{x} d x

Simplify

ln (x^{24}) : 24 ln (x)

= \int \frac{24 ln ( x )}{x} d x

Take the constant out:

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

= 24 \cdot \int \frac{ln ( x )}{x} d x

Apply u-substitution

= 24 \cdot \int u d u

Apply the Power Rule

= 24 \cdot \frac{u ^{2}}{2}

Substitute back

u = ln (x)

= 24 \cdot \frac{ln ^{2} ( x )}{2}

Simplify

24 \cdot \frac{ln ^{2} ( x )}{2} : 12 ln^{2} (x)

= 12 ln^{2} (x)

Add a constant to the solution

= 12 ln^{2} (x) + C

What is the integral of (ln(x^{24}))/x ?
The integral of (ln(x^{24}))/x is 12ln^2(x)+C