What is the integral of 1/(x\sqrt[3]{ln(x^5))} ?

The integral of 1/(x\sqrt[3]{ln(x^5))} is 3/(2\sqrt[3]{5)}ln^{2/3}(x)+C

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integral of 1/(x\sqrt[3]{ln(x^5))}

\int \frac{1}{x 3 ln ( x ^{5} )} d x

\frac{3}{2 3 5} ln^{\frac{2}{3}} (x) + C

Solution steps

\int \frac{1}{x 3 ln ( x ^{5} )} d x

3 ln (x^{5}) = 35 3 ln (x)

= \int \frac{1}{3 5 x 3 ln ( x )} d x

Take the constant out:

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

= \frac{1}{3 5} \cdot \int \frac{1}{x 3 ln ( x )} d x

Apply u-substitution

= \frac{1}{3 5} \cdot \int \frac{1}{3 u} d u

Apply the Power Rule

= \frac{1}{3 5} \cdot \frac{3}{2} u^{\frac{2}{3}}

Substitute back

u = ln (x)

= \frac{1}{3 5} \cdot \frac{3}{2} ln^{\frac{2}{3}} (x)

Simplify

\frac{1}{3 5} \cdot \frac{3}{2} ln^{\frac{2}{3}} (x) : \frac{3}{2 3 5} ln^{\frac{2}{3}} (x)

= \frac{3}{2 3 5} ln^{\frac{2}{3}} (x)

Add a constant to the solution

= \frac{3}{2 3 5} ln^{\frac{2}{3}} (x) + C

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What is the integral of 1/(x\sqrt[3]{ln(x^5))} ?
The integral of 1/(x\sqrt[3]{ln(x^5))} is 3/(2\sqrt[3]{5)}ln^{2/3}(x)+C