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

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

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

\int \frac{I n ∣ x ∣}{x} d x

\frac{2 I n x ^{\frac{3}{2}}}{3} + C

Solution steps

\int \frac{I n ∣ x ∣}{x} d x

Take the constant out:

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

= I n \cdot \int \frac{∣ x ∣}{x} d x

Apply u-substitution

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

Take the constant out:

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

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

Apply the Power Rule

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

Substitute back

u = x

= I n \cdot 2 \cdot \frac{( x ) ^{3}}{3}

Simplify

I n \cdot 2 \cdot \frac{( x ) ^{3}}{3} : \frac{2 I n x ^{\frac{3}{2}}}{3}

= \frac{2 I n x ^{\frac{3}{2}}}{3}

Add a constant to the solution

= \frac{2 I n x ^{\frac{3}{2}}}{3} + C

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