What is the integral of ((3-2ln(x))^{1/4})/x ?

The integral of ((3-2ln(x))^{1/4})/x is -2/5 (-2ln(x)+3)^{5/4}+C

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integral of ((3-2ln(x))^{1/4})/x

\int \frac{( 3 - 2 ln ( x ) ) ^{\frac{1}{4}}}{x} d x

- \frac{2}{5} (- 2 ln (x) + 3)^{\frac{5}{4}} + C

Solution steps

\int \frac{( 3 - 2 ln ( x ) ) ^{\frac{1}{4}}}{x} d x

Apply u-substitution

: \int - \frac{u ^{\frac{1}{4}}}{2} d u

= \int - \frac{u ^{\frac{1}{4}}}{2} d u

Take the constant out:

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

= - \frac{1}{2} \cdot \int u^{\frac{1}{4}} d u

Apply the Power Rule

: \frac{4}{5} u^{\frac{5}{4}}

= - \frac{1}{2} \cdot \frac{4}{5} u^{\frac{5}{4}}

Substitute back

u = 3 - 2 ln (x)

= - \frac{1}{2} \cdot \frac{4}{5} (3 - 2 ln (x))^{\frac{5}{4}}

Simplify

- \frac{1}{2} \cdot \frac{4}{5} (3 - 2 ln (x))^{\frac{5}{4}} : - \frac{2}{5} (- 2 ln (x) + 3)^{\frac{5}{4}}

= - \frac{2}{5} (- 2 ln (x) + 3)^{\frac{5}{4}}

Add a constant to the solution

= - \frac{2}{5} (- 2 ln (x) + 3)^{\frac{5}{4}} + C

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What is the integral of ((3-2ln(x))^{1/4})/x ?
The integral of ((3-2ln(x))^{1/4})/x is -2/5 (-2ln(x)+3)^{5/4}+C