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

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

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

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

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

Solution steps

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

Simplify

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

Apply u-substitution

: \int u^{\frac{3}{2}} d u

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

Apply the Power Rule

: \frac{2}{5} u^{\frac{5}{2}}

= \frac{2}{5} u^{\frac{5}{2}}

Substitute back

u = ln (x)

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

Add a constant to the solution

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

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