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

The integral of ((ln(x))^p)/x is (ln^{p+1}(x))/(p+1)+C

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

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

\frac{ln ^{p + 1} ( x )}{p + 1} + C

Solution steps

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

Simplify

= \int \frac{ln ^{p} ( x )}{x} d x

Apply u-substitution

: \int u^{p} d u

= \int u^{p} d u

Apply the Power Rule

: \frac{u ^{p + 1}}{p + 1}

= \frac{u ^{p + 1}}{p + 1}

Substitute back

u = ln (x)

= \frac{ln ^{p + 1} ( x )}{p + 1}

Add a constant to the solution

= \frac{ln ^{p + 1} ( x )}{p + 1} + C

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What is the integral of ((ln(x))^p)/x ?
The integral of ((ln(x))^p)/x is (ln^{p+1}(x))/(p+1)+C