What is the integral of ((ln(x))^5)/(5x) ?

The integral of ((ln(x))^5)/(5x) is 1/30 ln^6(x)+C

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integral of ((ln(x))^5)/(5x)

\int \frac{( ln ( x ) ) ^{5}}{5 x} d x

\frac{1}{30} ln^{6} (x) + C

Solution steps

\int \frac{( ln ( x ) ) ^{5}}{5 x} d x

Take the constant out:

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

= \frac{1}{5} \cdot \int \frac{( ln ( x ) ) ^{5}}{x} d x

Apply u-substitution

: \int u^{5} d u

= \frac{1}{5} \cdot \int u^{5} d u

Apply the Power Rule

: \frac{u ^{6}}{6}

= \frac{1}{5} \cdot \frac{u ^{6}}{6}

Substitute back

u = ln (x)

= \frac{1}{5} \cdot \frac{ln ^{6} ( x )}{6}

Simplify

\frac{1}{5} \cdot \frac{ln ^{6} ( x )}{6} : \frac{1}{30} ln^{6} (x)

= \frac{1}{30} ln^{6} (x)

Add a constant to the solution

= \frac{1}{30} ln^{6} (x) + C

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What is the integral of ((ln(x))^5)/(5x) ?
The integral of ((ln(x))^5)/(5x) is 1/30 ln^6(x)+C