What is the integral of 1/(x(2-3ln(x))^{2/3)} ?

The integral of 1/(x(2-3ln(x))^{2/3)} is -(2-3ln(x))^{1/3}+C

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integral of 1/(x(2-3ln(x))^{2/3)}

\int \frac{1}{x ( 2 - 3 ln ( x ) ) ^{\frac{2}{3}}} d x

- (2 - 3 ln (x))^{\frac{1}{3}} + C

Solution steps

\int \frac{1}{x ( 2 - 3 ln ( x ) ) ^{\frac{2}{3}}} d x

Apply u-substitution

= \int - \frac{1}{3 u ^{\frac{2}{3}}} d u

Take the constant out:

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

= - \frac{1}{3} \cdot \int \frac{1}{u ^{\frac{2}{3}}} d u

Apply the Power Rule

= - \frac{1}{3} \cdot 3 u^{\frac{1}{3}}

Substitute back

u = 2 - 3 ln (x)

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

Simplify

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

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

Add a constant to the solution

= - (2 - 3 ln (x))^{\frac{1}{3}} + C

What is the integral of 1/(x(2-3ln(x))^{2/3)} ?
The integral of 1/(x(2-3ln(x))^{2/3)} is -(2-3ln(x))^{1/3}+C