What is the integral of x*ln(1/x) ?

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

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integral of x*ln(1/x)

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

\int \frac{1 - e ^{- 4 x}}{1 + e ^{- 4 x}} d x

(1 + 4 x)^{\frac{3}{2}}

\frac{d P}{d t} = 4 P + a

\frac{d}{d x} (\frac{1}{8 x ^{- 4}} + \frac{1}{4 x ^{2}})

\frac{d}{d x} (2 x - cos (x))

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