What is the integral of (ln(x^{16}))/x ?

The integral of (ln(x^{16}))/x is 16ln(x)ln|x|-8ln|x|^2+C

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integral of (ln(x^{16}))/x

\int \frac{ln ( x ^{16} )}{x} d x

16 ln (x) ln ∣ x ∣ - 8 ln ∣ x ∣^{2} + C

Solution steps

\int \frac{ln ( x ^{16} )}{x} d x

Apply Integration By Parts

: ln (x^{16}) ln ∣ x ∣ - \int \frac{16 ln ∣ x ∣}{x} d x

= ln (x^{16}) ln ∣ x ∣ - \int \frac{16 ln ∣ x ∣}{x} d x

\int \frac{16 ln ∣ x ∣}{x} d x = 8 ln ∣ x ∣^{2}

= ln (x^{16}) ln ∣ x ∣ - 8 ln ∣ x ∣^{2}

Apply log rule

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x),

assuming

x \geq 0

= 16 ln (x) ln ∣ x ∣ - 8 ln ∣ x ∣^{2}

Add a constant to the solution

= 16 ln (x) ln ∣ x ∣ - 8 ln ∣ x ∣^{2} + C

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What is the integral of (ln(x^{16}))/x ?
The integral of (ln(x^{16}))/x is 16ln(x)ln|x|-8ln|x|^2+C