What is the integral of (ln(x^{21}))/x ?

The integral of (ln(x^{21}))/x is 21/2 ln^2(x)+C

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integral of (ln(x^{21}))/x

\int \frac{ln ( x ^{21} )}{x} d x

\frac{21}{2} ln^{2} (x) + C

Solution steps

\int \frac{ln ( x ^{21} )}{x} d x

Simplify

ln (x^{21}) : 21 ln (x)

= \int \frac{21 ln ( x )}{x} d x

Take the constant out:

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

= 21 \cdot \int \frac{ln ( x )}{x} d x

Apply u-substitution

: \int u d u

= 21 \cdot \int u d u

Apply the Power Rule

: \frac{u ^{2}}{2}

= 21 \cdot \frac{u ^{2}}{2}

Substitute back

u = ln (x)

= 21 \cdot \frac{ln ^{2} ( x )}{2}

Simplify

21 \cdot \frac{ln ^{2} ( x )}{2} : \frac{21}{2} ln^{2} (x)

= \frac{21}{2} ln^{2} (x)

Add a constant to the solution

= \frac{21}{2} ln^{2} (x) + C

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What is the integral of (ln(x^{21}))/x ?
The integral of (ln(x^{21}))/x is 21/2 ln^2(x)+C