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

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

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

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

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

Solution steps

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

Simplify

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

Apply u-substitution

= \int u^{n} d u

Apply the Power Rule

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

Substitute back

u = ln (x)

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

Add a constant to the solution

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

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