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

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

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

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

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

Solution steps

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

Apply log rule

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x),

assuming

x \geq 0

= \int \frac{1}{2} ln (x) d x

Take the constant out:

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

= \frac{1}{2} \cdot \int ln (x) d x

Apply Integration By Parts

: x ln (x) - \int 1 d x

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

\int 1 d x = x

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

Add a constant to the solution

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

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