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

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

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

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

2 x^{\frac{1}{2}} ln (x) - 4 x^{\frac{1}{2}} + C

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{2}{x ^{\frac{1}{2}}} d x = 4 x^{\frac{1}{2}}

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

Add a constant to the solution

= 2 x^{\frac{1}{2}} ln (x) - 4 x^{\frac{1}{2}} + C

x \to 0 lim ((1 + 4 x)^{\frac{5}{x}})

x \to \frac{π}{4} lim (sec (x) tan (x))

x \to 0 + lim (ln (sin (x)))

\frac{d}{d x} (- x^{3} + 4)

\frac{\partial}{\partial x} (\frac{y}{2 x} - 1)

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