What is the integral of (sqrt(ln|x|))/x ?

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

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integral of (sqrt(ln|x|))/x

\int \frac{ln ∣ x ∣}{x} d x

\frac{2}{3} ln ∣ x ∣^{\frac{3}{2}} + C

Solution steps

\int \frac{ln ∣ x ∣}{x} d x

Apply u-substitution

: \int 2 u^{2} d u

= \int 2 u^{2} d u

Take the constant out:

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

= 2 \cdot \int u^{2} d u

Apply the Power Rule

: \frac{u ^{3}}{3}

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

Substitute back

u = ln ∣ x ∣

= 2 \cdot \frac{( ln ∣ x ∣ ) ^{3}}{3}

Simplify

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

= \frac{2}{3} ln ∣ x ∣^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{3} ln ∣ x ∣^{\frac{3}{2}} + C

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\int 0.1 e^{- 0.1 x} d x

\frac{d y}{d t} = sin (3 t) - 2 y

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